Trigonometric identity

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

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}}