Trig Formulae

\(\sin^2 x + \cos^2 x = 1\)

\(\tan^2 x + 1 = \sec^2 x\)

\(1 + \cot^2 x = \csc^2 x\)

Summary

\(\sin^2 x + \cos^2 x = 1\)

\(\tan^2 x + 1 = \sec^2 x\)

\(1 + \cot^2 x = \csc^2 x\)

\(\sin(90^\circ - x) = \cos x\)

\(\cos(90^\circ - x) = \sin x\)

\(\tan(90^\circ - x) = \cot x\)

\(\sec(90^\circ - x) = \csc x\)

\(\csc(90^\circ - x) = \sec x\)

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

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

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

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

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

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

\(\sin 2x = 2\sin x\cos x\)

\(\cos 2x = \cos^2 x - \sin^2 x\)

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

\(\sin^2 \frac{x}{2} = \frac{1-\cos x}{2}\)

\(\cos^2 \frac{x}{2} = \frac{1+\cos x}{2}\)

\(\tan \frac{x}{2} = \frac{\sin x}{1+\cos x}\)

\(\sin x + \sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2}\)

\(\sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2}\)

\(\cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2}\)

\(\cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\)

\(\sin x\sin y = \frac{1}{2}[\cos(x-y)-\cos(x+y)]\)

\(\cos x\cos y = \frac{1}{2}[\cos(x-y)+\cos(x+y)]\)

\(\sin x\cos y = \frac{1}{2}[\sin(x+y)+\sin(x-y)]\)

\(\sin 3x = 3\sin x - 4\sin^3 x\)

\(\cos 3x = 4\cos^3 x - 3\cos x\)

\(\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}\)