A trigonometric identity is an equality that is true for every angle for which both sides are defined. Identities let different trigonometric expressions describe the same quantity.

The Pythagorean identity

Every point $(\cos\theta,\sin\theta)$ on the unit circle satisfies $x^2+y^2=1$. Therefore,

$$\sin^2\theta+\cos^2\theta=1.$$

This is the fundamental Pythagorean identity. Rearranging it gives, for example,

$$1-\sin^2\theta=\cos^2\theta.$$

Quotient and reciprocal identities

Tangent is related to sine and cosine by

$$\tan\theta=\frac{\sin\theta}{\cos\theta}.$$

The reciprocal functions are

$$\sec\theta=\frac1{\cos\theta},\qquad \csc\theta=\frac1{\sin\theta},\qquad \cot\theta=\frac{\cos\theta}{\sin\theta},$$

where the denominators are nonzero.

Combining angles

Sine and cosine of sums and differences can be expanded:

$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta,$$

$$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta.$$

Replacing $\beta$ by $-\beta$ gives the corresponding difference formulas. Setting $\alpha=\beta=\theta$ produces double-angle identities such as

$$\sin(2\theta)=2\sin\theta\cos\theta.$$

Using identities

An identity can be verified by starting from one side and applying known identities and valid algebraic transformations until the other side is obtained. Because trigonometric expressions may be undefined at some angles, domain restrictions remain part of the equality.