The Pythagorean triganometric identity

\[\sin^2\theta + \cos^2\theta = 1\]

Simple manipulation gives us the following additional identities …

\[\sin^2\theta = 1 -\cos^2\theta\] \[\sin\theta = \pm \sqrt{1 -\cos^2\theta}\] \[\cos^2\theta = 1 -\sin^2\theta\] \[\cos\theta = \pm \sqrt{1 -\sin^2\theta}\] \[\tan^2\theta = \frac{1}{\cos^2\theta} - 1 = sec^2\theta - 1\] \[\tan\theta = \pm \sqrt{sec^2\theta - 1}\]

Angle sum and difference identities

\[\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta\] \[\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta\] \[\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}\]

Double angle sum identities

These follow immediately from the angle sum identities.

\[\sin(2\theta) = \sin\theta\cos\theta + \cos\theta\sin\theta = 2\sin\theta\cos\theta\] \[\cos(2\theta) = \cos\theta\cos\theta - \sin\theta\sin\theta = \cos^2\theta - \sin^2\theta\] \[\tan(2\theta) = \frac{\tan\theta + \tan\theta}{1 - \tan\theta\tan\theta} = \frac{2tan\theta}{1 - \tan^2\theta}\]