Trigonometric Identities

Identities are equations that are true for all values of the variable.

Pythagorean Identities

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

Derived from this: $\tan^2\theta + 1 = \sec^2\theta$ $1 + \cot^2\theta = \csc^2\theta$

Quotient Identities

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

Reciprocal Identities

$\csc\theta = \frac{1}{\sin\theta}$ $\sec\theta = \frac{1}{\cos\theta}$ $\cot\theta = \frac{1}{\tan\theta}$

Even-Odd Identities

Cosine is even: $\cos(-\theta) = \cos\theta$ Sine is odd: $\sin(-\theta) = -\sin\theta$ Tangent is odd: $\tan(-\theta) = -\tan\theta$

Sum and Difference Formulas

$\sin(A + B) = \sin A \cos B + \cos A \sin B$ $\sin(A - B) = \sin A \cos B - \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$ $\cos(A - B) = \cos A \cos B + \sin A \sin B$

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

Double Angle Formulas

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

$\cos(2\theta) = \cos^2\theta - \sin^2\theta$ $= 2\cos^2\theta - 1$ $= 1 - 2\sin^2\theta$

$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Using Identities

Example: Simplify $\frac{\sin^2\theta}{1 + \cos\theta}$

$= \frac{1 - \cos^2\theta}{1 + \cos\theta}$ $= \frac{(1 - \cos\theta)(1 + \cos\theta)}{1 + \cos\theta}$ $= 1 - \cos\theta$