List of trigonometric identities

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Pythagorean identities

Main article: Pythagorean trigonometric identity

Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse. The triangle shaded blue illustrates the identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + \cot^2\theta = \csc^2\theta} , and the red triangle shows that $\tan ^{2}\theta +1=\sec ^{2}\theta$.

The basic relationship between the sine and cosine is given by the Pythagorean identity:

$$\sin ^{2}\theta +\cos ^{2}\theta =1,$$

where $\sin ^{2}\theta$ means $(\sin \theta )^{2}$ and $\cos ^{2}\theta$ means $(\cos \theta )^{2}.$

This can be viewed as a version of the Pythagorean theorem, and follows from the equation $x^{2}+y^{2}=1$ for the unit circle. This equation can be solved for either the sine or the cosine:

$${\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}$$

where the sign depends on the quadrant of $\theta .$

Dividing this identity by $\sin ^{2}\theta$, $\cos ^{2}\theta$, or both yields the following identities:

$${\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \\&1+\tan ^{2}\theta =\sec ^{2}\theta \\&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}$$

Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):

Each trigonometric function in terms of each of the other five.

in terms of	$\sin \theta$	$\csc \theta$	$\cos \theta$	$\sec \theta$	$\tan \theta$	$\cot \theta$
$\sin \theta =$	$\sin \theta$	${\frac {1}{\csc \theta }}$	$\pm {\sqrt {1-\cos ^{2}\theta }}$	$\pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}$	$\pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}$	$\pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}$
$\csc \theta =$	${\frac {1}{\sin \theta }}$	$\csc \theta$	$\pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}$	$\pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}$	$\pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}$	$\pm {\sqrt {1+\cot ^{2}\theta }}$
$\cos \theta =$	$\pm {\sqrt {1-\sin ^{2}\theta }}$	$\pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sec \theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{1}{\sqrt{1 + \tan^2 \theta}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{\cot \theta}{\sqrt{1 + \cot^2 \theta}}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec \theta =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{1}{\sqrt{1 - \sin^2 \theta}}}	$\pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\cos \theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\sqrt{1 + \tan^2 \theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{\sqrt{1 + \cot^2 \theta}}{\cot \theta}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan \theta =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{1}{\sqrt{\csc^2 \theta - 1}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{\sqrt{1 - \cos^2 \theta}}{\cos \theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\sqrt{\sec^2 \theta - 1}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\cot \theta}}
$\cot \theta =$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\sqrt{\csc^2 \theta - 1}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{\cos \theta}{\sqrt{1 - \cos^2 \theta}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\frac{1}{\sqrt{\sec^2 \theta - 1}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\tan \theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot \theta}

Reflections, shifts, and periodicity

By examining the unit circle, one can establish the following properties of the trigonometric functions.

Reflections

Transformation of coordinates (a,b) when shifting the reflection angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} in increments of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4}} .

When the direction of a Euclidean vector is represented by an angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta,} this is the angle determined by the free vector (starting at the origin) and the positive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis. If a line (vector) with direction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is reflected about a line with direction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha,} then the direction angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta^{\prime}} of this reflected line (vector) has the value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta^{\prime} = 2 \alpha - \theta.}

The values of the trigonometric functions of these angles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta,\;\theta^{\prime}} for specific angles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function. These are also known as reduction formulae.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} reflected in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0} odd/even identities	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} reflected in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{\pi}{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} reflected in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{\pi}{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} reflected in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{3\pi}{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} reflected in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \pi} compare to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(-\theta) = -\sin \theta}	$\sin \left({\tfrac {\pi }{2}}-\theta \right)=\cos \theta$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\pi - \theta) = +\sin \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin\left(\tfrac{3\pi}{2} - \theta\right) =-\cos \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(2\pi - \theta) = -\sin(\theta) = \sin(-\theta)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(-\theta) =+ \cos \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos\left(\tfrac{\pi}{2} - \theta\right) = \sin \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\pi - \theta) = -\cos \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos\left(\tfrac{3\pi}{2} - \theta\right) = -\sin \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(2\pi - \theta) = +\cos(\theta) = \cos(-\theta)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(-\theta) = -\tan \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan\left(\tfrac{\pi}{2} - \theta\right) = \cot \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\pi - \theta) = -\tan \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan\left(\tfrac{3\pi}{2} - \theta\right) = +\cot \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(2\pi - \theta) = -\tan(\theta) = \tan(-\theta)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(-\theta) = -\csc \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc\left(\tfrac{\pi}{2} - \theta\right) = \sec \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(\pi - \theta) =+ \csc \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc\left(\tfrac{3\pi}{2} - \theta\right) = -\sec \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(2\pi - \theta) = -\csc(\theta) = \csc(-\theta)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(-\theta) = +\sec \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec\left(\tfrac{\pi}{2} - \theta\right) = \csc \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(\pi - \theta) = -\sec \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec\left(\tfrac{3\pi}{2} - \theta\right) = -\csc \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(2\pi - \theta) = +\sec(\theta) = \sec(-\theta)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(-\theta) = -\cot \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot\left(\tfrac{\pi}{2} - \theta\right) = \tan \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(\pi - \theta) = -\cot \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot\left(\tfrac{3\pi}{2} - \theta\right) = +\tan \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(2\pi - \theta) = -\cot(\theta) = \cot(-\theta)}

Shifts and periodicity

Transformation of coordinates (a,b) when shifting the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} in increments of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{2}} .

Shift by one quarter period	Shift by one half period	Shift by full periods	Period
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\theta \pm \tfrac{\pi}{2}) = \pm\cos \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\theta + \pi) = -\sin \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\theta + k\cdot 2\pi) = +\sin \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\theta \pm \tfrac{\pi}{2}) = \mp\sin \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\theta + \pi) = -\cos \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\theta + k\cdot 2\pi) = +\cos \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(\theta \pm \tfrac{\pi}{2}) = \pm\sec \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(\theta + \pi) = -\csc \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(\theta + k\cdot 2\pi) = +\csc \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(\theta \pm \tfrac{\pi}{2}) = \mp\csc \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(\theta + \pi) = -\sec \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(\theta + k\cdot 2\pi) = +\sec \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\theta \pm \tfrac{\pi}{4}) = \tfrac{\tan \theta \pm 1}{1\mp \tan \theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\theta + \tfrac{\pi}{2}) = -\cot \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\theta + k\cdot \pi) = +\tan \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(\theta \pm \tfrac{\pi}{4}) = \tfrac{\cot \theta \mp 1}{1\pm \cot \theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(\theta + \tfrac{\pi}{2}) = -\tan\theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(\theta + k\cdot \pi) = +\cot \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi}

Signs

The sign of trigonometric functions depends on quadrant of the angle. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {-\pi} < \theta \leq \pi} and sgn is the sign function,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sgn(\sin \theta) = \sgn(\csc \theta) &= \begin{cases} +1 & \text{if}\ \ 0 < \theta < \pi \\ -1 & \text{if}\ \ {-\pi} < \theta < 0 \\ 0 & \text{if}\ \ \theta \in \{0, \pi \} \end{cases} \\[5mu] \sgn(\cos \theta) = \sgn(\sec \theta) &= \begin{cases} +1 & \text{if}\ \ {-\tfrac12\pi} < \theta < \tfrac12\pi \\ -1 & \text{if}\ \ {-\pi} < \theta < -\tfrac12\pi \ \ \text{or}\ \ \tfrac12\pi < \theta < \pi\\ 0 & \text{if}\ \ \theta \in \bigl\{{-\tfrac12\pi}, \tfrac12\pi \bigr\} \end{cases} \\[5mu] \sgn(\tan \theta) = \sgn(\cot \theta) &= \begin{cases} +1 & \text{if}\ \ {-\pi} < \theta < -\tfrac12\pi \ \ \text{or}\ \ 0 < \theta < \tfrac12\pi \\ -1 & \text{if}\ \ {-\tfrac12\pi} < \theta < 0 \ \ \text{or}\ \ \tfrac12\pi < \theta < \pi \\ 0 & \text{if}\ \ \theta \in \bigl\{{-\tfrac12\pi}, 0, \tfrac12\pi, \pi \bigr\} \end{cases} \end{align}}

The trigonometric functions are periodic with common period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi,} so for values of θ outside the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({-\pi}, \pi],} they take repeating values (see § Shifts and periodicity above).

Angle sum and difference identities

See also: Proofs of trigonometric identities § Angle sum identities, and Small-angle approximation § Angle sum and difference

Illustration of angle addition formulae for the sine and cosine of acute angles. Emphasized segment is of unit length.

Diagram showing the angle difference identities for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\alpha - \beta)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\alpha - \beta)} .

These are also known as the angle addition and subtraction theorems (or formulae). Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta \\ \sin(\alpha - \beta) &= \sin \alpha \cos \beta - \cos \alpha \sin \beta \\ \cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \cos(\alpha - \beta) &= \cos \alpha \cos \beta + \sin \alpha \sin \beta \end{align}}

The angle difference identities for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\alpha - \beta)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\alpha - \beta)} can be derived from the angle sum versions by substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\beta} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} and using the facts that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(-\beta) = -\sin(\beta)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(-\beta) = \cos(\beta)} . They can also be derived by using a slightly modified version of the figure for the angle sum identities, both of which are shown here.

These identities are summarized in the first two rows of the following table, which also includes sum and difference identities for the other trigonometric functions.

Sine	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\alpha \pm \beta)}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta}
Cosine	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(\alpha \pm \beta)}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta}
Tangent	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(\alpha \pm \beta)}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}}
Cosecant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(\alpha \pm \beta)}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sec \alpha \sec \beta \csc \alpha \csc \beta}{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta}}
Secant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(\alpha \pm \beta)}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\sec \alpha \sec \beta \csc \alpha \csc \beta}{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta}}
Cotangent	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(\alpha \pm \beta)}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	${\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}$
Arcsine	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arcsin x \pm \arcsin y}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arcsin\left(x\sqrt{1-y^2} \pm y\sqrt{1-x^2\vphantom{y}}\right)}
Arccosine	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arccos x \pm \arccos y}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arccos\left(xy \mp \sqrt{\left(1-x^2\right)\left(1-y^2\right)}\right)}
Arctangent	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan x \pm \arctan y}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan\left(\frac{x \pm y}{1 \mp xy}\right)}
Arccotangent	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arccot x \pm \arccot y}			Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arccot\left(\frac{xy \mp 1}{y \pm x}\right)}

Sines and cosines of sums of infinitely many angles

When the series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sum_{i=1}^\infty \theta_i} converges absolutely then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\sin}\biggl(\sum_{i=1}^\infty \theta_i\biggl) &= \sum_{\text{odd}\ k \ge 1} (-1)^\frac{k-1}{2} \!\! \sum_{\begin{smallmatrix} A \subseteq \{\,1,2,3,\dots\,\} \\ \left|A\right| = k\end{smallmatrix}} \biggl(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\biggr) \\ {\cos}\biggl(\sum_{i=1}^\infty \theta_i\biggr) &= \sum_{\text{even}\ k \ge 0} (-1)^\frac{k}{2} \, \sum_{\begin{smallmatrix} A \subseteq \{\,1,2,3,\dots\,\} \\ \left|A\right| = k\end{smallmatrix}} \biggl(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\biggr) . \end{align}}

Because the series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sum_{i=1}^\infty \theta_i} converges absolutely, it is necessarily the case that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{i \to \infty} \theta_i = 0,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{i \to \infty} \sin \theta_i = 0,} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{i \to \infty} \cos \theta_i = 1.} In particular, in these two identities an asymmetry appears that is not seen in the case of sums of finitely many angles: in each product, there are only finitely many sine factors but there are cofinitely many cosine factors. Terms with infinitely many sine factors would necessarily be equal to zero.

When only finitely many of the angles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_i} are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.

Tangents and cotangents of sums

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_k} (for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = 0, 1, 2, 3, \ldots} ) be the kth-degree elementary symmetric polynomial in the variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i = \tan \theta_i} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 0, 1, 2, 3, \ldots,} that is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} e_0 &= 1 \\[6pt] e_1 &= \sum_i x_i &&= \sum_i \tan\theta_i \\[6pt] e_2 &= \sum_{i<j} x_i x_j &&= \sum_{i<j} \tan\theta_i \tan\theta_j \\[6pt] e_3 &= \sum_{i<j<k} x_i x_j x_k &&= \sum_{i<j<k} \tan\theta_i \tan\theta_j \tan\theta_k \\ &\ \ \vdots &&\ \ \vdots \end{align}}

Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\tan}\Bigl(\sum_i \theta_i\Bigr) &= \frac{{\sin}\bigl(\sum_i \theta_i\bigr) / \prod_i \cos \theta_i} {{\cos}\bigl(\sum_i \theta_i\bigr) / \prod_i \cos \theta_i} \\[10pt] &= \frac {\displaystyle \sum_{\text{odd}\ k \ge 1} (-1)^\frac{k-1}{2} \sum_{ \begin{smallmatrix} A \subseteq \{1,2,3,\dots\} \\ \left|A\right| = k\end{smallmatrix}} \prod_{i \in A} \tan\theta_i} {\displaystyle \sum_{\text{even}\ k \ge 0} ~ (-1)^\frac{k}{2} ~~ \sum_{ \begin{smallmatrix} A \subseteq \{1,2,3,\dots\} \\ \left|A\right| = k\end{smallmatrix}} \prod_{i \in A} \tan\theta_i} = \frac{e_1 - e_3 + e_5 -\cdots}{e_0 - e_2 + e_4 - \cdots} \\[10pt] {\cot}\Bigl(\sum_i \theta_i\Bigr) &= \frac{e_0 - e_2 + e_4 - \cdots}{e_1 - e_3 + e_5 -\cdots} \end{align}}

using the sine and cosine sum formulae above.

The number of terms on the right side depends on the number of terms on the left side.

For example: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \tan(\theta_1 + \theta_2) & = \frac{ e_1 }{ e_0 - e_2 } = \frac{ x_1 + x_2 }{ 1 \ - \ x_1 x_2 } = \frac{ \tan\theta_1 + \tan\theta_2 }{ 1 \ - \ \tan\theta_1 \tan\theta_2 }, \\[8pt] \tan(\theta_1 + \theta_2 + \theta_3) & = \frac{ e_1 - e_3 }{ e_0 - e_2 } = \frac{ (x_1 + x_2 + x_3) \ - \ (x_1 x_2 x_3) }{ 1 \ - \ (x_1x_2 + x_1 x_3 + x_2 x_3) }, \\[8pt] \tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) & = \frac{ e_1 - e_3 }{ e_0 - e_2 + e_4 } \\[8pt] & = \frac{ (x_1 + x_2 + x_3 + x_4) \ - \ (x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4) }{ 1 \ - \ (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) \ + \ (x_1 x_2 x_3 x_4) }, \end{align}}

and so on. The case of only finitely many terms can be proved by mathematical induction. The case of infinitely many terms can be proved by using some elementary inequalities.

Secants and cosecants of sums

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\sec}\Bigl(\sum_i \theta_i \Bigr) &= \frac{\prod_i \sec\theta_i}{e_0 - e_2 + e_4 - \cdots} \\[8pt] {\csc}\Bigl(\sum_i \theta_i \Bigr) &= \frac{\prod_i \sec\theta_i }{e_1 - e_3 + e_5 - \cdots} \end{align}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e_k} is the kth-degree elementary symmetric polynomial in the n variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i = \tan \theta_i,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 1, \ldots, n,} and the number of terms in the denominator and the number of factors in the product in the numerator depend on the number of terms in the sum on the left. The case of only finitely many terms can be proved by mathematical induction on the number of such terms.

For example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sec(\alpha+\beta+\gamma) &= \frac{\sec\alpha \sec\beta \sec\gamma} {1 - \tan\alpha\tan\beta - \tan\alpha\tan\gamma - \tan\beta\tan\gamma} \\[8pt] \csc(\alpha+\beta+\gamma) &= \frac{\sec\alpha \sec\beta \sec\gamma} {\tan\alpha + \tan\beta + \tan\gamma - \tan\alpha\tan\beta\tan\gamma}. \end{align}}

Ptolemy's theorem

Main article: Ptolemy's theorem

See also: History of trigonometry § Classical antiquity

Diagram illustrating the relation between Ptolemy's theorem and the angle sum trig identity for sine. Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.

Ptolemy's theorem is important in the history of trigonometric identities, as it is how results equivalent to the sum and difference formulas for sine and cosine were first proved. It states that in a cyclic quadrilateral Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ABCD} , as shown in the accompanying figure, the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. In the special cases of one of the diagonals or sides being a diameter of the circle, this theorem gives rise directly to the angle sum and difference trigonometric identities. The relationship follows most easily when the circle is constructed to have a diameter of length one, as shown here.

By Thales's theorem, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \angle DAB} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \angle DCB} are both right angles. The right-angled triangles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle DAB} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle DCB} both share the hypotenuse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{BD}} of length 1. Thus, the side Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AB} = \sin \alpha} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AD} = \cos \alpha} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{BC} = \sin \beta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{CD} = \cos \beta} .

By the inscribed angle theorem, the central angle subtended by the chord Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AC}} at the circle's center is twice the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \angle ADC} , i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2(\alpha + \beta)} . Therefore, the symmetrical pair of red triangles each has the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha + \beta} at the center. Each of these triangles has a hypotenuse of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac{1}{2}} , so the length of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AC}} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle 2 \times \frac{1}{2} \sin(\alpha + \beta)} , i.e. simply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\alpha + \beta)} . The quadrilateral's other diagonal is the diameter of length 1, so the product of the diagonals' lengths is also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\alpha + \beta)} .

When these values are substituted into the statement of Ptolemy's theorem that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\overline{AC}|\cdot |\overline{BD}|=|\overline{AB}|\cdot |\overline{CD}|+|\overline{AD}|\cdot |\overline{BC}|} , this yields the angle sum trigonometric identity for sine: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta } . The angle difference formula for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(\alpha - \beta)} can be similarly derived by letting the side Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{CD}} serve as a diameter instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{BD}} .

Multiple-angle and half-angle formulae

Tn is the nth Chebyshev polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos (n\theta) = T_n (\cos \theta )}
de Moivre's formula, i is the imaginary unit	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos (n\theta) +i\sin (n\theta)=(\cos \theta +i\sin \theta)^n}

Multiple-angle formulae

Double-angle formulae

Visual demonstration of the double-angle formula for sine. For the above isosceles triangle with unit sides and angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\theta} , the area ⁠1/2⁠ × base × height is calculated in two orientations. When upright, the area is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta \cos \theta} . When on its side, the same area is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac{1}{2} \sin 2\theta} . Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin 2\theta = 2 \sin \theta \cos \theta.}

Formulae for twice an angle.

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin (2\theta) = 2 \sin \theta \cos \theta = (\sin \theta +\cos \theta)^2 - 1 = \frac{2 \tan \theta} {1 + \tan^2 \theta}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos (2\theta) = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta = \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan (2\theta) = \frac{2 \tan \theta} {1 - \tan^2 \theta}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot (2\theta) = \frac{\cot^2 \theta - 1}{2 \cot \theta} = \frac{1 - \tan^2 \theta} {2 \tan \theta}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec (2\theta) = \frac{\sec^2 \theta}{2 - \sec^2 \theta} = \frac{1 + \tan^2 \theta} {1 - \tan^2 \theta}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc (2\theta) = \frac{\sec \theta \csc \theta}{2} = \frac{1 + \tan^2 \theta} {2 \tan \theta}}

Triple-angle formulae

Formulae for triple angles.

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin (3\theta) =3\sin\theta - 4\sin^3\theta = 4\sin\theta\sin\left(\frac{\pi}{3} -\theta\right)\sin\left(\frac{\pi}{3} + \theta\right)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos (3\theta) = 4 \cos^3\theta - 3 \cos\theta =4\cos\theta\cos\left(\frac{\pi}{3} -\theta\right)\cos\left(\frac{\pi}{3} + \theta\right)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan (3\theta) = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta} = \tan \theta\tan\left(\frac{\pi}{3} - \theta\right)\tan\left(\frac{\pi}{3} + \theta\right)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot (3\theta) = \frac{3 \cot\theta - \cot^3\theta}{1 - 3 \cot^2\theta}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec (3\theta) = \frac{\sec^3\theta}{4-3\sec^2\theta}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc (3\theta) = \frac{\csc^3\theta}{3\csc^2\theta-4}}

Multiple-angle formulae

Formulae for multiple angles.

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sin(n\theta) &= \sum_{k\text{ odd}} (-1)^\frac{k-1}{2} {n \choose k}\cos^{n-k} \theta \sin^k \theta = \sin\theta\sum_{i=0}^{(n+1)/2}\sum_{j=0}^{i} (-1)^{i-j} {n \choose 2i + 1}{i \choose j} \cos^{n-2(i-j)-1} \theta \\ {}&=2^{(n-1)} \prod_{k=0}^{n-1} \sin(k\pi/n+\theta) \end{align}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(n\theta) = \sum_{k\text{ even}} (-1)^\frac{k}{2} {n \choose k}\cos^{n-k} \theta \sin^k \theta = \sum_{i=0}^{n/2}\sum_{j=0}^{i} (-1)^{i-j} {n \choose 2i}{i \choose j} \cos^{n-2(i-j)} \theta }
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos((2n+1)\theta)=(-1)^n 2^{2n}\prod_{k=0}^{2n}\cos(k\pi/(2n+1)-\theta)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(2 n \theta)=(-1)^n 2^{2n-1} \prod_{k=0}^{2n-1} \cos((1+2k)\pi/(4n)-\theta)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(n\theta) = \frac{\sum_{k\text{ odd}} (-1)^\frac{k-1}{2} {n \choose k}\tan^k \theta}{\sum_{k\text{ even}} (-1)^\frac{k}{2} {n \choose k}\tan^k \theta}}

Chebyshev method

The Chebyshev method is a recursive algorithm for finding the nth multiple angle formula knowing the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n-1)} th and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n-2)} th values.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(nx)} can be computed from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos((n-1)x)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos((n-2)x)} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(x)} with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(nx)=2 \cos x \cos((n-1)x) - \cos((n-2)x).}

This can be proved by adding together the formulae

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cos ((n-1)x + x) &= \cos ((n-1)x) \cos x-\sin ((n-1)x) \sin x \\ \cos ((n-1)x - x) &= \cos ((n-1)x) \cos x+\sin ((n-1)x) \sin x \end{align}}

It follows by induction that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(nx)} is a polynomial of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x,} the so-called Chebyshev polynomial of the first kind, see Chebyshev polynomials#Trigonometric definition.

Similarly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(nx)} can be computed from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin((n-1)x),} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin((n-2)x),} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(nx)=2 \cos x \sin((n-1)x)-\sin((n-2)x)} This can be proved by adding formulae for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin((n-1)x+x)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin((n-1)x-x).}

Serving a purpose similar to that of the Chebyshev method, for the tangent we can write:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan (nx) = \frac{\tan ((n-1)x) + \tan x}{1- \tan ((n-1)x) \tan x}\,.}

Half-angle formulae

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sin \frac{\theta}{2} &= \sgn\left(\sin\frac\theta2\right) \sqrt{\frac{1 - \cos \theta}{2}} \\[3pt] \cos \frac{\theta}{2} &= \sgn\left(\cos\frac\theta2\right) \sqrt{\frac{1 + \cos\theta}{2}} \\[3pt] \tan \frac{\theta}{2} &= \frac{1 - \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 + \cos \theta} = \csc \theta - \cot \theta = \frac{\tan\theta}{1 + \sec{\theta}} \\[6mu] &= \sgn(\sin \theta) \sqrt\frac{1 - \cos \theta}{1 + \cos \theta} = \frac{-1 + \sgn(\cos \theta) \sqrt{1+\tan^2\theta}}{\tan\theta} \\[3pt] \cot \frac{\theta}{2} &= \frac{1 + \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 - \cos \theta} = \csc \theta + \cot \theta = \sgn(\sin \theta) \sqrt\frac{1 + \cos \theta}{1 - \cos \theta} \\ \sec \frac{\theta}{2} &= \sgn\left(\cos\frac\theta2\right) \sqrt{\frac{2}{1 + \cos\theta}} \\ \csc \frac{\theta}{2} &= \sgn\left(\sin\frac\theta2\right) \sqrt{\frac{2}{1 - \cos\theta}} \\ \end{align}}

Also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \tan\frac{\eta\pm\theta}{2} &= \frac{\sin\eta \pm \sin\theta}{\cos\eta + \cos\theta} \\[3pt] \tan\left(\frac{\theta}{2} + \frac{\pi}{4}\right) &= \sec\theta + \tan\theta \\[3pt] \sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}} &= \frac{\left|1 - \tan\frac{\theta}{2}\right|}{\left|1 + \tan\frac{\theta}{2}\right|} \end{align}}

Table

See also: Tangent half-angle formula

These can be shown by using either the sum and difference identities or the multiple-angle formulae.

	Sine	Cosine	Tangent	Cotangent
Double-angle formula	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sin (2\theta) &= 2 \sin \theta \cos \theta \ \\ &= \frac{2 \tan \theta} {1 + \tan^2 \theta} \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cos (2\theta) &= \cos^2 \theta - \sin^2 \theta \\ &= 2 \cos^2 \theta - 1 \\ &= 1 - 2 \sin^2 \theta \\ &= \frac{1 - \tan^2 \theta} {1 + \tan^2 \theta} \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan (2\theta) = \frac{2 \tan \theta} {1 - \tan^2 \theta}}	$\cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}$
Triple-angle formula	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sin (3\theta) &= - \sin^3\theta + 3 \cos^2\theta \sin\theta\\ &= - 4\sin^3\theta + 3\sin\theta \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cos (3\theta) &= \cos^3\theta - 3 \sin^2 \theta\cos \theta \\ &= 4 \cos^3\theta - 3 \cos\theta \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan (3\theta) = \frac{3 \tan\theta - \tan^3\theta}{1 - 3 \tan^2\theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot (3\theta) = \frac{3 \cot\theta - \cot^3\theta}{1 - 3 \cot^2\theta}}
Half-angle formula	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} &\sin \frac{\theta}{2} = \sgn\left(\sin\frac\theta2\right) \sqrt{\frac{1 - \cos \theta}{2}} \\ \\ &\left(\text{or }\sin^2\frac{\theta}{2} = \frac{1 - \cos\theta}{2}\right) \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} &\cos \frac{\theta}{2} = \sgn\left(\cos\frac\theta2\right) \sqrt{\frac{1 + \cos\theta}{2}} \\ \\ &\left(\text{or } \cos^2\frac{\theta}{2} = \frac{1 + \cos\theta}{2}\right) \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \tan \frac{\theta}{2} &= \csc \theta - \cot \theta \\ &= \pm\, \sqrt\frac{1 - \cos \theta}{1 + \cos \theta} \\[3pt] &= \frac{\sin \theta}{1 + \cos \theta} \\[3pt] &= \frac{1 - \cos \theta}{\sin \theta} \\[5pt] \tan\frac{\eta + \theta}{2} &= \frac{\sin\eta + \sin\theta}{\cos\eta + \cos\theta} \\[5pt] \tan\left(\frac{\theta}{2} + \frac{\pi}{4}\right) &= \sec\theta + \tan\theta \\[5pt] \sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}} &= \frac{\left|1 - \tan\frac{\theta}{2}\right|}{\left|1 + \tan\frac{\theta}{2}\right|} \\[5pt] \tan\frac{\theta}{2} &= \frac{\tan\theta}{1 + \sqrt{1 + \tan^2\theta}} \\ &\text{for } \theta \in \left(-\tfrac{\pi}{2},\tfrac{\pi}{2} \right) \end{align}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cot \frac{\theta}{2} &= \csc \theta + \cot \theta \\ &= \pm\, \sqrt\frac{1 + \cos \theta}{1 - \cos \theta} \\[3pt] &= \frac{\sin \theta}{1 - \cos \theta} \\[4pt] &= \frac{1 + \cos \theta}{\sin \theta} \end{align}}

The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools, by field theory.

A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x³ − 3x + d = 0, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. However, the discriminant of this equation is positive, so this equation has three real roots (of which only one is the solution for the cosine of the one-third angle). None of these solutions are reducible to a real algebraic expression, as they use intermediate complex numbers under the cube roots.

Power-reduction formulae

Obtained by solving the second and third versions of the cosine double-angle formula.

Sine	Cosine	Other
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2\theta = \frac{1 - \cos (2\theta)}{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2\theta = \frac{1 + \cos (2\theta)}{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2\theta \cos^2\theta = \frac{1 - \cos (4\theta)}{8}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^3\theta = \frac{3 \sin\theta - \sin (3\theta)}{4}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^3\theta = \frac{3 \cos\theta + \cos (3\theta)}{4}}	$\sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin(2\theta )-\sin(6\theta )}{32}}$
$\sin ^{4}\theta ={\frac {3-4\cos(2\theta )+\cos(4\theta )}{8}}$	$\cos ^{4}\theta ={\frac {3+4\cos(2\theta )+\cos(4\theta )}{8}}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^4\theta \cos^4\theta = \frac{3-4\cos (4\theta) + \cos (8\theta)}{128}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^5\theta = \frac{10 \sin\theta - 5 \sin (3\theta) + \sin (5\theta)}{16}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^5\theta = \frac{10 \cos\theta + 5 \cos (3\theta) + \cos (5\theta)}{16}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^5\theta \cos^5\theta = \frac{10\sin (2\theta) - 5\sin (6\theta) + \sin (10\theta)}{512}}

Cosine power-reduction formula: an illustrative diagram. The red, orange and blue triangles are all similar, and the red and orange triangles are congruent. The hypotenuse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AD}} of the blue triangle has length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \cos \theta} . The angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \angle DAE} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} , so the base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AE}} of that triangle has length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \cos^2 \theta} . That length is also equal to the summed lengths of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{BD}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AF}} , i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + \cos (2\theta)} . Therefore, $2\cos ^{2}\theta =1+\cos(2\theta )$. Dividing both sides by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} yields the power-reduction formula for cosine: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^2\theta =} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac{1}{2}(1 + \cos (2\theta)) } . The half-angle formula for cosine can be obtained by replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta/2} and taking the square-root of both sides: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \cos \left(\theta / 2\right) = \pm \sqrt{\left(1 + \cos \theta\right) /2}.}

Sine power-reduction formula: an illustrative diagram. The shaded blue and green triangles, and the red-outlined triangle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle EBD} are all right-angled and similar, and all contain the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} . The hypotenuse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{BD}} of the red-outlined triangle has length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \sin \theta} , so its side Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{DE}} has length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \sin^2 \theta} . The line segment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AE}} has length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos 2 \theta} and sum of the lengths of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AE}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{DE}} equals the length of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{AD}} , which is 1. Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos 2 \theta + 2 \sin^2 \theta = 1 } . Subtracting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos 2 \theta} from both sides and dividing by 2 by two yields the power-reduction formula for sine: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2 \theta = } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac{1}{2} (1 - \cos (2\theta))} . The half-angle formula for sine can be obtained by replacing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta/2} and taking the square-root of both sides: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \sin \left(\theta/2\right) = \pm \sqrt{\left(1 - \cos \theta\right)/2}.} Note that this figure also illustrates, in the vertical line segment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{EB}} , that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin 2 \theta = 2 \sin \theta \cos \theta} .

In general terms of powers of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \theta} the following is true, and can be deduced using De Moivre's formula, Euler's formula and the binomial theorem.

if n is ...	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^n \theta}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^n \theta}
n is odd	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} \cos{\big((n-2k)\theta\big)}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} (-1)^{\left(\frac{n-1}{2}-k\right)} \binom{n}{k} \sin{\big((n-2k)\theta\big)}}
n is even	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos^n\theta = \frac{1}{2^n} \binom{n}{\frac{n}{2}} + \frac{2}{2^n} \sum_{k=0}^{\frac{n}{2}-1} \binom{n}{k} \cos{\big((n-2k)\theta\big)}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^n\theta = \frac{1}{2^n} \binom{n}{\frac{n}{2}} + \frac{2}{2^n} \sum_{k=0}^{\frac{n}{2}-1} (-1)^{\left(\frac{n}{2}-k\right)} \binom{n}{k} \cos{\big((n-2k)\theta\big)}}

Product-to-sum and sum-to-product identities

Proof of the sum-and-difference-to-product cosine identity for prosthaphaeresis calculations using an isosceles triangle

The product-to-sum identities or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations. See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.

Product-to-sum identities

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \theta\, \cos \varphi = {\cos(\theta - \varphi) + \cos(\theta + \varphi) \over 2}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta\, \sin \varphi = {\cos(\theta - \varphi) - \cos(\theta + \varphi) \over 2}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta\, \cos \varphi = {\sin(\theta + \varphi) + \sin(\theta - \varphi) \over 2}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \theta\, \sin \varphi = {\sin(\theta + \varphi) - \sin(\theta - \varphi) \over 2}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan \theta\, \tan \varphi =\frac{\cos(\theta-\varphi)-\cos(\theta+\varphi)}{\cos(\theta-\varphi)+\cos(\theta+\varphi)}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan \theta\, \cot \varphi = \frac{\sin(\theta + \varphi) + \sin(\theta - \varphi)}{\sin(\theta + \varphi) - \sin(\theta - \varphi)}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \prod_{k=1}^n \cos \theta_k & = \frac{1}{2^n}\sum_{e\in S} \cos(e_1\theta_1+\cdots+e_n\theta_n) \\[6pt] & \text{where }e = (e_1,\ldots,e_n) \in S=\{1,-1\}^n \end{align}}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{k=1}^n \sin\theta_k=\frac{(-1)^{\left\lfloor\frac {n}{2}\right\rfloor}}{2^n}\begin{cases} \displaystyle\sum_{e\in S}\cos(e_1\theta_1+\cdots+e_n\theta_n)\prod_{j=1}^n e_j \;\text{if}\; n\; \text{is even},\\ \displaystyle\sum_{e\in S}\sin(e_1\theta_1+\cdots+e_n\theta_n)\prod_{j=1}^n e_j \;\text{if}\; n\; \text{is odd} \end{cases}}

Sum-to-product identities

Diagram illustrating sum-to-product identities for sine and cosine. The blue right-angled triangle has angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} and the red right-angled triangle has angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} . Both have a hypotenuse of length 1. Auxiliary angles, here called Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} , are constructed such that $p=(\theta +\varphi )/2$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=(\theta-\varphi)/2} . Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta = p+q } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi = p-q } . This allows the two congruent purple-outline triangles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AFG} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle FCE} to be constructed, each with hypotenuse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos q} and angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} at their base. The sum of the heights of the red and blue triangles is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta + \sin \varphi} , and this is equal to twice the height of one purple triangle, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \sin p \cos q} . Writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} in that equation in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} yields a sum-to-product identity for sine: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta + \sin \varphi = 2 \sin\left( \frac{\theta + \varphi}{2} \right) \cos\left( \frac{\theta - \varphi}{2} \right)} . Similarly, the sum of the widths of the red and blue triangles yields the corresponding identity for cosine.

The sum-to-product identities are as follows:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta \pm \sin \varphi = 2 \sin\left( \frac{\theta \pm \varphi}{2} \right) \cos\left( \frac{\theta \mp \varphi}{2} \right)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \theta + \cos \varphi = 2 \cos\left( \frac{\theta + \varphi} {2} \right) \cos\left( \frac{\theta - \varphi}{2} \right)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \theta - \cos \varphi = -2\sin\left( \frac{\theta + \varphi}{2}\right) \sin\left(\frac{\theta - \varphi}{2}\right)}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan\theta\pm\tan\varphi=\frac{\sin(\theta\pm \varphi)}{\cos\theta\,\cos\varphi}}

Hermite's cotangent identity

Main article: Hermite's cotangent identity

Charles Hermite demonstrated the following identity. Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1, \ldots, a_n} are complex numbers, no two of which differ by an integer multiple of π. Let

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{n,k} = \prod_{\begin{smallmatrix} 1 \le j \le n \\ j \neq k \end{smallmatrix}} \cot(a_k - a_j)}

(in particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{1,1},} being an empty product, is 1). Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(z - a_1)\cdots\cot(z - a_n) = \cos\frac{n\pi}{2} + \sum_{k=1}^n A_{n,k} \cot(z - a_k).}

The simplest non-trivial example is the case n = 2:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(z - a_1)\cot(z - a_2) = -1 + \cot(a_1 - a_2)\cot(z - a_1) + \cot(a_2 - a_1)\cot(z - a_2).}

Finite products of trigonometric functions

For coprime integers n, m

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{k=1}^n \left(2a + 2\cos\left(\frac{2 \pi k m}{n} + x\right)\right) = 2\left( T_n(a)+{(-1)}^{n+m}\cos(n x) \right)}

where T_n is the Chebyshev polynomial.

The following relationship holds for the sine function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}.}

More generally for an integer n > 0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(nx) = 2^{n-1}\prod_{k=0}^{n-1} \sin\left(\frac{k}{n}\pi + x\right) = 2^{n-1}\prod_{k=1}^{n} \sin\left(\frac{k}{n}\pi - x\right).}

or written in terms of the chord function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \operatorname{crd}x \equiv 2\sin\tfrac12x} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{crd}(nx) = \prod_{k=1}^{n} \operatorname{crd}\left(\frac{k}{n}2\pi - x\right).}

This comes from the factorization of the polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle z^n - 1} into linear factors (cf. root of unity): For any complex z and an integer n > 0,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^n - 1 = \prod_{k=1}^{n}\left( z - \exp\Bigl(\frac{k}{n}2\pi i\Bigr)\right).}

Linear combinations

For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} .

Sine and cosine

The linear combination, or harmonic addition, of sine and cosine waves is equivalent to a single sine wave with a phase shift and scaled amplitude,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\cos x+b\sin x=c\cos(x+\varphi)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} and $\varphi$ are defined as so:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} c &= \sgn(a) \sqrt{a^2 + b^2}, \\ \varphi &= {\arctan}\bigl({-b/a}\bigr), \end{align}}

given that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \neq 0.}

Arbitrary phase shift

More generally, for arbitrary phase shifts, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \sin(x + \theta_a) + b \sin(x + \theta_b)= c \sin(x+\varphi)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} satisfy:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} c^2 &= a^2 + b^2 + 2ab\cos \left(\theta_a - \theta_b \right) , \\ \tan \varphi &= \frac{a \sin \theta_a + b \sin \theta_b}{a \cos \theta_a + b \cos \theta_b}. \end{align}}

More than two sinusoids

See also: Phasor addition

The general case reads

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_i a_i \sin(x + \theta_i) = a \sin(x + \theta),} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2 = \sum_{i,j}a_i a_j \cos(\theta_i - \theta_j)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan\theta = \frac{\sum_i a_i \sin\theta_i}{\sum_i a_i \cos\theta_i}.}

Lagrange's trigonometric identities

These identities, named after Joseph Louis Lagrange, are: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sum_{k=0}^n \sin k\theta & = \frac{\cos \tfrac12\theta - \cos\left(\left(n + \tfrac12\right)\theta\right)}{2\sin\tfrac12\theta}\\[5pt] \sum_{k=0}^n \cos k\theta & = \frac{\sin \tfrac12\theta + \sin\left(\left(n + \tfrac12\right)\theta\right)}{2\sin\tfrac12\theta} \end{align}} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \not\equiv 0 \pmod{2\pi}.}

A related function is the Dirichlet kernel:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_n(\theta) = 1 + 2\sum_{k=1}^n \cos k\theta = \frac{\sin\left(\left(n + \tfrac12 \right)\theta\right)}{\sin \tfrac12 \theta}.}

A similar identity is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=1}^n \cos (2k -1)\alpha = \frac{\sin (2n \alpha)}{2 \sin \alpha}.}

The proof is the following. By using the angle sum and difference identities, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin (A + B) - \sin (A - B) = 2 \cos A \sin B.} Then let's examine the following formula,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \sin \alpha \sum_{k=1}^n \cos (2k - 1)\alpha = 2\sin \alpha \cos \alpha + 2 \sin \alpha \cos 3\alpha + 2 \sin \alpha \cos 5 \alpha + \ldots + 2 \sin \alpha \cos (2n - 1) \alpha } and this formula can be written by using the above identity,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} & 2 \sin \alpha \sum_{k=1}^n \cos (2k - 1)\alpha \\ &\quad= \sum_{k=1}^n (\sin (2k \alpha) - \sin (2(k - 1)\alpha)) \\ &\quad= (\sin 2\alpha - \sin 0) + (\sin 4 \alpha - \sin 2 \alpha) + (\sin 6 \alpha - \sin 4 \alpha) + \ldots + (\sin (2n \alpha) - \sin (2(n - 1) \alpha)) \\ &\quad= \sin (2n \alpha). \end{align}}

So, dividing this formula with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \sin \alpha} completes the proof.

Certain linear fractional transformations

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is given by the linear fractional transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \frac{(\cos\alpha)x - \sin\alpha}{(\sin\alpha)x + \cos\alpha},} and similarly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = \frac{(\cos\beta)x - \sin\beta}{(\sin\beta)x + \cos\beta},} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\big(g(x)\big) = g\big(f(x)\big) = \frac{\big(\cos(\alpha+\beta)\big)x - \sin(\alpha+\beta)}{\big(\sin(\alpha+\beta)\big)x + \cos(\alpha+\beta)}.}

More tersely stated, if for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} we let $f_{\alpha }$ be what we called Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} above, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_\alpha \circ f_\beta = f_{\alpha+\beta}.}

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the slope of a line, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is the slope of its rotation through an angle of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \alpha.}

Relation to the complex exponential function

Main article: Euler's formula

Euler's formula states that, for any real number x: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{ix} = \cos x + i\sin x,} where i is the imaginary unit. Substituting −x for x gives us: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{-ix} = \cos(-x) + i\sin(-x) = \cos x - i\sin x.}

These two equations can be used to solve for cosine and sine in terms of the exponential function. Specifically, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x = \frac{e^{ix} + e^{-ix}}{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x = \frac{e^{ix} - e^{-ix}}{2i}}

These formulae are useful for proving many other trigonometric identities. For example, that e^i(θ+φ) = e^iθ e^iφ means that

cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).

That the real part of the left hand side equals the real part of the right hand side is an angle addition formula for cosine. The equality of the imaginary parts gives an angle addition formula for sine.

The following table expresses the trigonometric functions and their inverses in terms of the exponential function and the complex logarithm.

Function	Inverse function
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arcsin x = -i\, \ln \left(ix + \sqrt{1 - x^2}\right)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arccos x = -i\ln\left(x+\sqrt{x^2-1}\right)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan \theta = -i\, \frac{e^{i\theta} - e^{-i\theta}}{e^{i\theta} + e^{-i\theta}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan x = \frac{i}{2} \ln \left(\frac{i + x}{i - x}\right)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc \theta = \frac{2i}{e^{i\theta} - e^{-i\theta}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arccsc x = -i\, \ln \left(\frac{i}{x} + \sqrt{1 - \frac{1}{x^2}}\right)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec \theta = \frac{2}{e^{i\theta} + e^{-i\theta}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arcsec x = -i\, \ln \left(\frac{1}{x} +i \sqrt{1 - \frac{1}{x^2}}\right)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot \theta = i\, \frac{e^{i\theta} + e^{-i\theta}}{e^{i\theta} - e^{-i\theta}}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arccot x = \frac{i}{2} \ln \left(\frac{x - i}{x + i}\right)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{cis} \theta = e^{i\theta}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{arccis} x = -i \ln x}

Series expansion

When using a power series expansion to define trigonometric functions, the following identities are obtained:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{(2n+1)!},} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n}}{(2n)!}.}

Infinite product formulae

For applications to special functions, the following infinite product formulae for trigonometric functions are useful:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sin x &= x \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2 n^2}\right), & \cos x &= \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2\left(n - \frac{1}{2}\right)\!\vphantom)^2}\right), \\[10mu] \sinh x &= x \prod_{n = 1}^\infty\left(1 + \frac{x^2}{\pi^2 n^2}\right), & \cosh x &= \prod_{n = 1}^\infty\left(1 + \frac{x^2}{\pi^2\left(n - \frac{1}{2}\right)\!\vphantom)^2}\right). \end{align}}

Inverse trigonometric functions

Main article: Inverse trigonometric functions

The following identities give the result of composing a trigonometric function with an inverse trigonometric function.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sin(\arcsin x) &=x & \cos(\arcsin x) &=\sqrt{1-x^2} & \tan(\arcsin x) &=\frac{x}{\sqrt{1 - x^2}} \\ \sin(\arccos x) &=\sqrt{1-x^2} & \cos(\arccos x) &=x & \tan(\arccos x) &=\frac{\sqrt{1 - x^2}}{x} \\ \sin(\arctan x) &=\frac{x}{\sqrt{1+x^2}} & \cos(\arctan x) &=\frac{1}{\sqrt{1+x^2}} & \tan(\arctan x) &=x \\ \sin(\arccsc x) &=\frac{1}{x} & \cos(\arccsc x) &=\frac{\sqrt{x^2 - 1}}{x} & \tan(\arccsc x) &=\frac{1}{\sqrt{x^2 - 1}} \\ \sin(\arcsec x) &=\frac{\sqrt{x^2 - 1}}{x} & \cos(\arcsec x) &=\frac{1}{x} & \tan(\arcsec x) &=\sqrt{x^2 - 1} \\ \sin(\arccot x) &=\frac{1}{\sqrt{1+x^2}} & \cos(\arccot x) &=\frac{x}{\sqrt{1+x^2}} & \tan(\arccot x) &=\frac{1}{x} \\ \end{align} }

Taking the multiplicative inverse of both sides of the each equation above results in the equations for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc = \frac{1}{\sin}, \;\sec = \frac{1}{\cos}, \text{ and } \cot = \frac{1}{\tan}.} The right hand side of the formula above will always be flipped. For example, the equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(\arcsin x)} is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cot(\arcsin x) = \frac{1}{\tan(\arcsin x)} = \frac{1}{\frac{x}{\sqrt{1 - x^2}}} = \frac{\sqrt{1 - x^2}}{x}} while the equations for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(\arccos x)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sec(\arccos x)} are: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \csc(\arccos x) = \frac{1}{\sin(\arccos x)} = \frac{1}{\sqrt{1-x^2}} \qquad \text{ and }\quad \sec(\arccos x) = \frac{1}{\cos(\arccos x)} = \frac{1}{x}.}

The following identities are implied by the reflection identities. They hold whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, r, s, -x, -r, \text{ and } -s} are in the domains of the relevant functions. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{9} \frac{\pi}{2} ~&=~ \arcsin(x) &&+ \arccos(x) ~&&=~ \arctan(r) &&+ \arccot(r) ~&&=~ \arcsec(s) &&+ \arccsc(s) \\[0.4ex] \pi ~&=~ \arccos(x) &&+ \arccos(-x) ~&&=~ \arccot(r) &&+ \arccot(-r) ~&&=~ \arcsec(s) &&+ \arcsec(-s) \\[0.4ex] 0 ~&=~ \arcsin(x) &&+ \arcsin(-x) ~&&=~ \arctan(r) &&+ \arctan(-r) ~&&=~ \arccsc(s) &&+ \arccsc(-s) \\[1.0ex] \end{alignat}}

Also, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \arctan x + \arctan \dfrac{1}{x} &= \begin{cases} \frac{\pi}{2}, & \text{if } x > 0 \\ - \frac{\pi}{2}, & \text{if } x < 0 \end{cases} \\ \arccot x + \arccot \dfrac{1}{x} &= \begin{cases} \frac{\pi}{2}, & \text{if } x > 0 \\ \frac{3\pi}{2}, & \text{if } x < 0 \end{cases} \\ \end{align}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arccos \frac{1}{x} = \arcsec x \qquad \text{ and } \qquad \arcsec \frac{1}{x} = \arccos x} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arcsin \frac{1}{x} = \arccsc x \qquad \text{ and } \qquad \arccsc \frac{1}{x} = \arcsin x}

The arctangent function can be expanded as a series: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan(nx) = \sum_{m = 1}^n \arctan\frac{x}{1 + (m - 1)mx^2} }

Identities without variables

In terms of the arctangent function we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arctan \frac{1}{2} = \arctan \frac{1}{3} + \arctan \frac{1}{7}.}

The curious identity known as Morrie's law, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos 20^\circ\cdot\cos 40^\circ\cdot\cos 80^\circ = \frac{1}{8},}

is a special case of an identity that contains one variable: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{j=0}^{k-1}\cos\left(2^j x\right) = \frac{\sin\left(2^k x\right)}{2^k\sin x}.}

Similarly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin 20^\circ\cdot\sin 40^\circ\cdot\sin 80^\circ = \frac{\sqrt{3}}{8}} is a special case of an identity with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 20^\circ} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x \cdot \sin \left(60^\circ - x\right) \cdot \sin \left(60^\circ + x\right) = \frac{\sin 3x}{4}.}

For the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 15^\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sin 15^\circ\cdot\sin 45^\circ\cdot\sin 75^\circ &= \frac{\sqrt{2}}{8}, \\ \sin 15^\circ\cdot\sin 75^\circ &= \frac{1}{4}. \end{align}}

For the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 10^\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin 10^\circ\cdot\sin 50^\circ\cdot\sin 70^\circ = \frac{1}{8}.}

The same cosine identity is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x \cdot \cos \left(60^\circ - x\right) \cdot \cos \left(60^\circ + x\right) = \frac{\cos 3x}{4}.}

Similarly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \cos 10^\circ\cdot\cos 50^\circ\cdot\cos 70^\circ &= \frac{\sqrt{3}}{8}, \\ \cos 15^\circ\cdot\cos 45^\circ\cdot\cos 75^\circ &= \frac{\sqrt{2}}{8}, \\ \cos 15^\circ\cdot\cos 75^\circ &= \frac{1}{4}. \end{align}}

Similarly, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \tan 50^\circ\cdot\tan 60^\circ\cdot\tan 70^\circ &= \tan 80^\circ, \\ \tan 40^\circ\cdot\tan 30^\circ\cdot\tan 20^\circ &= \tan 10^\circ. \end{align}}

The following is perhaps not as readily generalized to an identity containing variables (but see explanation below): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos 24^\circ + \cos 48^\circ + \cos 96^\circ + \cos 168^\circ = \frac{1}{2}.}

Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \frac{2\pi}{21} + \cos\left(2\cdot\frac{2\pi}{21}\right) + \cos\left(4\cdot\frac{2\pi}{21}\right) + \cos\left( 5\cdot\frac{2\pi}{21}\right) + \cos\left( 8\cdot\frac{2\pi}{21}\right) + \cos\left(10\cdot\frac{2\pi}{21}\right) = \frac{1}{2}.}

The factors 1, 2, 4, 5, 8, 10 may start to make the pattern clear: they are those integers less than ⁠21/2⁠ that are relatively prime to (or have no prime factors in common with) 21. The last several examples are corollaries of a basic fact about the irreducible cyclotomic polynomials: the cosines are the real parts of the zeroes of those polynomials; the sum of the zeroes is the Möbius function evaluated at (in the very last case above) 21; only half of the zeroes are present above. The two identities preceding this last one arise in the same fashion with 21 replaced by 10 and 15, respectively.

Other cosine identities include: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 2\cos \frac{\pi}{3} &= 1, \\ 2\cos \frac{\pi}{5} \times 2\cos \frac{2\pi}{5} &= 1, \\ 2\cos \frac{\pi}{7} \times 2\cos \frac{2\pi}{7}\times 2\cos \frac{3\pi}{7} &= 1, \end{align}} and so forth for all odd numbers, and hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos \frac{\pi}{3}+\cos \frac{\pi}{5} \times \cos \frac{2\pi}{5} + \cos \frac{\pi}{7} \times \cos \frac{2\pi}{7} \times \cos \frac{3\pi}{7} + \dots = 1.}

Many of those curious identities stem from more general facts like the following: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{k=1}^{n-1} \sin\frac{k\pi}{n} = \frac{n}{2^{n-1}}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{k=1}^{n-1} \cos\frac{k\pi}{n} = \frac{\sin\frac{\pi n}{2}}{2^{n-1}}.}

Combining these gives us Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{k=1}^{n-1} \tan\frac{k\pi}{n} = \frac{n}{\sin\frac{\pi n}{2}}}

If n is an odd number (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 2 m + 1} ) we can make use of the symmetries to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{k=1}^{m} \tan\frac{k\pi}{2m+1} = \sqrt{2m+1}}

The transfer function of the Butterworth low pass filter can be expressed in terms of polynomial and poles. By setting the frequency as the cutoff frequency, the following identity can be proved: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \prod_{k=1}^n \sin\frac{\left(2k - 1\right)\pi}{4n} = \prod_{k=1}^{n} \cos\frac{\left(2k-1\right)\pi}{4n} = \frac{\sqrt{2}}{2^n}}

Computing π

An efficient way to compute π to a large number of digits is based on the following identity without variables, due to Machin. This is known as a Machin-like formula: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239}} or, alternatively, by using an identity of Leonhard Euler: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4} = 5 \arctan\frac{1}{7} + 2 \arctan\frac{3}{79}} or by using Pythagorean triples: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi = \arccos\frac{4}{5} + \arccos\frac{5}{13} + \arccos\frac{16}{65} = \arcsin\frac{3}{5} + \arcsin\frac{12}{13} + \arcsin\frac{63}{65}.}

Others include: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3},} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi = \arctan 1 + \arctan 2 + \arctan 3,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4} = 2\arctan \frac{1}{3} + \arctan \frac{1}{7}.}

Generally, for numbers t₁, ..., t_n−1 ∈ (−1, 1) for which θ_n = Σⁿ⁻¹
_k=1 arctan t_k ∈ (π/4, 3π/4), let t_n = tan(π/2 − θ_n) = cot θ_n. This last expression can be computed directly using the formula for the cotangent of a sum of angles whose tangents are t₁, ..., t_n−1 and its value will be in (−1, 1). In particular, the computed t_n will be rational whenever all the t₁, ..., t_n−1 values are rational. With these values, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\pi}{2} & = \sum_{k=1}^n \arctan(t_k) \\ \pi & = \sum_{k=1}^n \sgn(t_k) \arccos\left(\frac{1 - t_k^2}{1 + t_k^2}\right) \\ \pi & = \sum_{k=1}^n \arcsin\left(\frac{2t_k}{1 + t_k^2}\right) \\ \pi & = \sum_{k=1}^n \arctan\left(\frac{2t_k}{1 - t_k^2}\right)\,, \end{align}}

where in all but the first expression, we have used tangent half-angle formulae. The first two formulae work even if one or more of the t_k values is not within (−1, 1). Note that if t = p/q is rational, then the (2t, 1 − t², 1 + t²) values in the above formulae are proportional to the Pythagorean triple (2pq, q² − p², q² + p²).

For example, for n = 3 terms, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{2} = \arctan\left(\frac{a}{b}\right) + \arctan\left(\frac{c}{d}\right) + \arctan\left(\frac{bd - ac}{ad + bc}\right)} for any a, b, c, d > 0.

An identity of Euclid

Euclid showed in Book XIII, Proposition 10 of his Elements that the area of the square on the side of a regular pentagon inscribed in a circle is equal to the sum of the areas of the squares on the sides of the regular hexagon and the regular decagon inscribed in the same circle. In the language of modern trigonometry, this says: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin^2 18^\circ + \sin^2 30^\circ = \sin^2 36^\circ.}

Ptolemy used this proposition to compute some angles in his table of chords in Book I, chapter 11 of Almagest.

Composition of trigonometric functions

These identities involve a trigonometric function of a trigonometric function:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(t \sin x) = J_0(t) + 2 \sum_{k=1}^\infty J_{2k}(t) \cos(2kx)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(t \sin x) = 2 \sum_{k=0}^\infty J_{2k+1}(t) \sin\big((2k+1)x\big)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(t \cos x) = J_0(t) + 2 \sum_{k=1}^\infty (-1)^kJ_{2k}(t) \cos(2kx)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(t \cos x) = 2 \sum_{k=0}^\infty(-1)^k J_{2k+1}(t) \cos\big((2k+1)x\big)}

where J_i are Bessel functions.

Further "conditional" identities for the case α + β + γ = 180°

A conditional trigonometric identity is a trigonometric identity that holds if specified conditions on the arguments to the trigonometric functions are satisfied. The following formulae apply to arbitrary plane triangles and follow from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha + \beta + \gamma = 180^{\circ},} as long as the functions occurring in the formulae are well-defined (the latter applies only to the formulae in which tangents and cotangents occur). Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \tan \alpha + \tan \beta + \tan \gamma &= \tan \alpha \tan \beta \tan \gamma \\ 1 &= \cot \beta \cot \gamma + \cot \gamma \cot \alpha + \cot \alpha \cot \beta \\ \cot\left(\frac{\alpha}{2}\right) + \cot\left(\frac{\beta}{2}\right) + \cot\left(\frac{\gamma}{2}\right) &= \cot\left(\frac{\alpha}{2}\right) \cot \left(\frac{\beta}{2}\right) \cot\left(\frac{\gamma}{2}\right) \\ 1 &= \tan\left(\frac{\beta}{2}\right)\tan\left(\frac{\gamma}{2}\right) + \tan\left(\frac{\gamma}{2}\right)\tan\left(\frac{\alpha}{2}\right) + \tan\left(\frac{\alpha}{2}\right)\tan\left(\frac{\beta}{2}\right) \\ \sin \alpha + \sin \beta + \sin \gamma &= 4\cos\left(\frac{\alpha}{2}\right)\cos\left(\frac{\beta}{2}\right)\cos\left(\frac{\gamma}{2}\right) \\ -\sin \alpha + \sin \beta + \sin \gamma &= 4\cos\left(\frac{\alpha}{2}\right)\sin\left(\frac{\beta}{2}\right)\sin\left(\frac{\gamma}{2}\right) \\ \cos \alpha + \cos \beta + \cos \gamma &= 4\sin\left(\frac{\alpha}{2}\right)\sin\left(\frac{\beta}{2}\right)\sin \left(\frac{\gamma}{2}\right) + 1 \\ -\cos \alpha + \cos \beta + \cos \gamma &= 4\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\beta}{2}\right)\cos \left(\frac{\gamma}{2}\right) - 1 \\ \sin (2\alpha) + \sin (2\beta) + \sin (2\gamma) &= 4\sin \alpha \sin \beta \sin \gamma \\ -\sin (2\alpha) + \sin (2\beta) + \sin (2\gamma) &= 4\sin \alpha \cos \beta \cos \gamma \\ \cos (2\alpha) + \cos (2\beta) + \cos (2\gamma) &= -4\cos \alpha \cos \beta \cos \gamma - 1 \\ -\cos (2\alpha) + \cos (2\beta) + \cos (2\gamma) &= -4\cos \alpha \sin \beta \sin \gamma + 1 \\ \sin^2\alpha + \sin^2\beta + \sin^2\gamma &= 2 \cos \alpha \cos \beta \cos \gamma + 2 \\ -\sin^2\alpha + \sin^2\beta + \sin^2\gamma &= 2 \cos \alpha \sin \beta \sin \gamma \\ \cos^2\alpha + \cos^2\beta + \cos^2\gamma &= -2 \cos \alpha \cos \beta \cos \gamma + 1 \\ -\cos^2\alpha + \cos^2\beta + \cos^2\gamma &= -2 \cos \alpha \sin \beta \sin \gamma + 1 \\ \sin^2 (2\alpha) + \sin^2 (2\beta) + \sin^2 (2\gamma) &= -2\cos (2\alpha) \cos (2\beta) \cos (2\gamma)+2 \\ \cos^2 (2\alpha) + \cos^2 (2\beta) + \cos^2 (2\gamma) &= 2\cos (2\alpha) \,\cos (2\beta) \,\cos (2\gamma) + 1 \\ 1 &= \sin^2 \left(\frac{\alpha}{2}\right) + \sin^2 \left(\frac{\beta}{2}\right) + \sin^2 \left(\frac{\gamma}{2}\right) + 2\sin \left(\frac{\alpha}{2}\right) \,\sin \left(\frac{\beta}{2}\right) \,\sin \left(\frac{\gamma}{2}\right) \end{align}}

Historical shorthands

Main articles: Versine and Exsecant

The versine, coversine, haversine, and exsecant were used in navigation. For example, the haversine formula was used to calculate the distance between two points on a sphere. They are rarely used today.

Miscellaneous

Dirichlet kernel

Main article: Dirichlet kernel

The Dirichlet kernel D_n(x) is the function occurring on both sides of the next identity: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + 2\cos x + 2\cos(2x) + 2\cos(3x) + \cdots + 2\cos(nx) = \frac{\sin\left(\left(n + \frac{1}{2}\right)x\right) }{\sin\left(\frac{1}{2}x\right)}.}

The convolution of any integrable function of period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \pi} with the Dirichlet kernel coincides with the function's Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} th-degree Fourier approximation. The same holds for any measure or generalized function.

Tangent half-angle substitution

Main article: Tangent half-angle substitution

If we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = \tan\frac x 2,} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x = \frac{2t}{1 + t^2};\qquad \cos x = \frac{1 - t^2}{1 + t^2};\qquad e^{i x} = \frac{1 + i t}{1 - i t}; \qquad dx = \frac{2\,dt}{1+t^2}, } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i x} = \cos x + i \sin x,} sometimes abbreviated to cis x.

When this substitution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} for tan ⁠x/2⁠ is used in calculus, it follows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x} is replaced by ⁠2t/1 + t²⁠, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x} is replaced by ⁠1 − t²/1 + t²⁠ and the differential dx is replaced by ⁠2 dt/1 + t²⁠. Thereby one converts rational functions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x} to rational functions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} in order to find their antiderivatives.

Viète's infinite product

See also: Viète's formula and Sinc function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos\frac{\theta}{2} \cdot \cos \frac{\theta}{4} \cdot \cos \frac{\theta}{8} \cdots = \prod_{n=1}^\infty \cos \frac{\theta}{2^n} = \frac{\sin \theta}{\theta} = \operatorname{sinc} \theta.}

See also

• Aristarchus's inequality
• Derivatives of trigonometric functions
• Exact trigonometric values (values of sine and cosine expressed in surds)
• Exsecant
• Half-side formula
• Hyperbolic function
• Laws for solution of triangles:
  • Law of cosines
    • Spherical law of cosines
  • Law of sines
  • Law of tangents
  • Law of cotangents
  • Mollweide's formula
• List of integrals of trigonometric functions
• Mnemonics in trigonometry
• Pentagramma mirificum
• Proofs of trigonometric identities
• Prosthaphaeresis
• Pythagorean theorem
• Tangent half-angle formula
• Trigonometric number
• Trigonometry
• Uses of trigonometry
• Versine and haversine

Bibliography

• Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications. ISBN 978-0-486-61272-0.
• Nielsen, Kaj L. (1966), Logarithmic and Trigonometric Tables to Five Places (2nd ed.), New York: Barnes & Noble, LCCN 61-9103
• Selby, Samuel M., ed. (1970), Standard Mathematical Tables (18th ed.), The Chemical Rubber Co.

External links

• Values of sin and cos, expressed in surds, for integer multiples of 3° and of ⁠5+5/8⁠°, and for the same angles csc and sec and tan
• Complete List of Trigonometric Formulas