Trigonometric Identities

Even Identity for sin

\sin(\theta) = -\sin(90 - \theta)

Odd Identity for sin

\sin(-\theta) = -\sin(\theta)

Odd Identity for csc

\csc(-\theta) = -\csc(\theta)

Odd Identity for tan

\tan(-\theta) = -\tan(\theta)

Odd Identity for cot

\cot(-\theta) = -\cot(\theta)

Cofunction Identity for csc

\csc(90 - \theta) = \sec(\theta)

Cofunction Identity for sec

\sec(90 - \theta) = \csc(\theta)

Cofunction Identity for tan

\tan(90 - \theta) = \cot(\theta)

Cofunction Identity for cot

\cot(90 - \theta) = \tan(\theta)

Sum Formula for sin

\sin(u + v) = \sin(u)\cos(v) + \cos(u)\sin(v)

Difference Formula for cos

\cos(u - v) = \cos(u)\cos(v) + \sin(u)\sin(v)

Even Identity for cos

\cos(-\theta) = \cos(\theta)

Even Identity for sec

\sec(-\theta) = \sec(\theta)

Cofunction Identity (sin and cos)

\sin(\theta) = \cos(90^\circ - \theta),\ \cos(\theta) = \sin(90^\circ - \theta)

Cofunction Identity (csc and sec)

\csc(\theta) = \sec(90^\circ - \theta),\ \sec(\theta) = \csc(90^\circ - \theta)

Cofunction Identity (tan and cot)

\tan(\theta) = \cot(90^\circ - \theta),\ \cot(\theta) = \tan(90^\circ - \theta)

Sum Formula (sin)

\sin(u + v) = \sin(u)\cos(v) + \cos(u)\sin(v)

Difference Formula (sin)

\sin(u - v) = \sin(u)\cos(v) - \cos(u)\sin(v)

Sum Formula (cos)

\cos(u + v) = \cos(u)\cos(v) - \sin(u)\sin(v)

Difference Formula (cos)

\cos(u - v) = \cos(u)\cos(v) + \sin(u)\sin(v)

Sum Formula (tan)

\tan(u + v) = \frac{\tan(u) + \tan(v)}{1 - \tan(u)\tan(v)}

Difference Formula (tan)

\tan(u - v) = \frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)}

Double Angle Formula (sin)

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

Double Angle Formula (cos)

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

Double Angle Formula (tan)

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

Half-Angle Formula (sin)

\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}

Half-Angle Formula (cos)

\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}

Half-Angle Formula (tan)

\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)}

Product-to-Sum (sin)

\sin(u)\sin(v) = \tfrac{1}{2}[\cos(u - v) - \cos(u + v)]

Product-to-Sum (cos)

\cos(u)\cos(v) = \tfrac{1}{2}[\cos(u - v) + \cos(u + v)]

Product-to-Sum (sin and cos)

\sin(u)\cos(v) = \tfrac{1}{2}[\sin(u + v) + \sin(u - v)]

Sum-to-Product (sin)

\sin(u) + \sin(v) = 2\sin\left(\frac{u + v}{2}\right)\cos\left(\frac{u - v}{2}\right)

Sum-to-Product (sin difference)

\sin(u) - \sin(v) = 2\cos\left(\frac{u + v}{2}\right)\sin\left(\frac{u - v}{2}\right)

Sum-to-Product (cos)

\cos(u) + \cos(v) = 2\cos\left(\frac{u + v}{2}\right)\cos\left(\frac{u - v}{2}\right)

Sum-to-Product (cos difference)

\cos(u) - \cos(v) = -2\sin\left(\frac{u + v}{2}\right)\sin\left(\frac{u - v}{2}\right)