Trigonometric Identities and Equations

An identity is an equation true for every permitted value of the variable, whereas an equation is true only for particular values that we solve for. Mastering identities lets us simplify expressions and rewrite equations into a solvable form. This lesson also covers the higher-level (HL) compound and double-angle identities and the inverse trigonometric functions.

The Pythagorean identity. The most important identity follows directly from the unit circle:

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

Dividing through by $\cos^2\theta$ gives $\tan^2\theta+1=\sec^2\theta$, and dividing by $\sin^2\theta$ gives $1+\cot^2\theta=\csc^2\theta$ (relevant when the reciprocal ratios appear). These let you convert between sine and cosine, for example $\cos\theta=\pm\sqrt{1-\sin^2\theta}$, with the sign chosen by