Trigonometric Identities

See also Hyperbolic Identities.

Let $\theta$ (theta), $\alpha$ (alpha), and $\beta$ (beta) be arbitrary angles (see Greek letters).

Basic Relationships

$$\begin{array}{rlr}\tan \theta & =\frac{\sin \theta }{\cos \theta }=\frac{1}{\cot \theta }& \text{(tangent)}\\ \cot \theta & =\frac{\cos \theta }{\sin \theta }=\frac{1}{\tan \theta }& \text{(cotangent)}\\ \sec \theta & =\frac{1}{\cos \theta }& \text{(secant)}\\ \csc \theta & =\frac{1}{\sec \theta }& \text{(cosecant)}\end{array}$$

Pythagorean Identities

$$\begin{array}{c}{\sin }^2\theta +{\cos }^2\theta =1\\ {\tan }^2\theta +1={\sec }^2\theta \\ {\cot }^2\theta +1={\csc }^2\theta \end{array}$$

Odd/Even Identities

$$\begin{array}{rlrlr}& \sin (-\theta )& & =-\sin \theta & \\ & \cos (-\theta )& & =\cos \theta & \\ & \tan (-\theta )& & =-\tan \theta & \\ & \sec (-\theta )& & =\sec \theta & \\ & \csc (-\theta )& & =-\csc \theta & \\ & \cot (-\theta )& & =-\cot \theta & \end{array}$$

Cofunction (Compliment) Identities

$$\begin{array}{r}\sin (90^{\circ }-\theta )=\cos \theta \\ \cos (90^{\circ }-\theta )=\sin \theta \\ \tan (90^{\circ }-\theta )=\cot \theta \\ \cot (90^{\circ }-\theta )=\tan \theta \\ \sec (90^{\circ }-\theta )=\csc \theta \\ \csc (90^{\circ }-\theta )=\sec \theta \end{array}$$

Sum and Difference Formulas

$$\begin{array}{rl}\sin (\alpha \pm \beta )& =\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \cos (\alpha \pm \beta )& =\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\ \tan (\alpha \pm \beta )& =\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }\end{array}$$

Note

The notation $\mp$ indicates that the sign here is taken to be opposite to the sign of $\pm$. For clarity, the formulas above really are a condensed version of the following six formulas.

$$\{\begin{array}{l}\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\ \sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\ \cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\ \sin (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta \\ \tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\\ \tan (\alpha -\beta )=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }\end{array}$$

Double-Angle Formulas

$$\begin{array}{rl}\sin 2\theta & =2\sin \theta \cos \theta \\ \cos 2\theta & ={\cos }^2\theta -{\sin }^2\theta \\ & =1-2{\sin }^2\theta \\ & =2{\cos }^2\theta -1\\ \tan 2\theta & =\frac{2\tan \theta }{1-{\tan }^2\theta }\end{array}$$

Half-Angle Formulas

$$\begin{array}{c}\sin \frac{\theta }{2}=\pm \sqrt{\frac{1-\cos \theta }{2}}\\ \cos \frac{\theta }{2}=\pm \sqrt{\frac{1+\cos \theta }{2}}\\ \tan \frac{\theta }{2}=\pm \sqrt{\frac{1-\cos \theta }{1+\cos \theta }}\end{array}$$