Trigonometric Identities

Trigonometric identities are equations involving trig functions that hold for all valid angles.

Core identities every student must memorise:

Pythagorean: $\sin^2\theta + \cos^2\theta = 1$, $1 + \tan^2\theta = \sec^2\theta$, $1 + \cot^2\theta = \csc^2\theta$.

Reciprocal: $\csc = 1/\sin$, $\sec = 1/\cos$, $\cot = 1/\tan$.

Quotient: $\tan\theta = \sin\theta / \cos\theta$.

Even-odd: $\sin(-\theta) = -\sin\theta$, $\cos(-\theta) = \cos\theta$.

Sum: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$.

Double-angle: $\sin(2\theta) = 2\sin\theta\cos\theta$, $\cos(2\theta) = \cos^2\theta - \sin^2\theta$.

For full reference see Trigonometry Identities Cheat Sheet. Identities power calculus integrals, Fourier series, and geometric proofs.