Table of Trigonometric Identities

A few of the more common trigonometric identities are given below.

Pythagorean Identities

• $\sin^2 x + \cos^2 x = 1$.

• $1 + \tan^2 x = \sec^2 x$.

• $1 + \cot^2 x = \csc^2 x$.

Even/Odd Identities

• $\sin (-x) = - \sin x$.

• $\cos (-x) = \cos x$.

• $\tan (-x) = -\tan x$.

Double Angle Identities

• $\sin (2x) = 2 \sin x \cos x$.

• $\cos (2x) = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x$.

• $\tan (2x) = \frac{2 \tan x}{1 - \tan^2 x}$.

Sum and Difference Identities

• $\sin (x + y) = \sin x \cos y + \cos x \sin y$.

• $\sin (x - y) = \sin x \cos y - \cos x \sin y$.

• $\cos (x + y) = \cos x \cos y - \sin x \sin y$.

• $\cos (x - y) = \cos x \cos y + \sin x \sin y$.

Reduction Identities

• $\sin^2 x = \frac{1 - \cos 2x}{2}$.

• $\cos^2 x = \frac{1 + \cos 2x}{2}$.