Trigonometric Identities

Pythagorean

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

Double Angle

$\sin 2\theta = 2\sin\theta\cos\theta$

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

Compound Angle

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$

$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

Work on one side only and transform it to match the other.

Example

Prove $\frac{\sin 2\theta}{1 + \cos 2\theta} = \tan\theta$.

LHS = $\frac{2\sin\theta\cos\theta}{1 + 2\cos^2\theta - 1} = \frac{2\sin\theta\cos\theta}{2\cos^2\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta$ = RHS ✓