List of Identities

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Reciprocal Identities

$$\begin{array}{rlrlrl}\csc (\theta )& ={\displaystyle \frac{1}{\sin (\theta )}}& \sec (\theta )& ={\displaystyle \frac{1}{\cos (\theta )}}& \cot (\theta )& ={\displaystyle \frac{1}{\tan (\theta )}}\\ \sin (\theta )& ={\displaystyle \frac{1}{\csc (\theta )}}& \cos (\theta )& ={\displaystyle \frac{1}{\sec (\theta )}}& \tan (\theta )& ={\displaystyle \frac{1}{\cot (\theta )}}\end{array}$$

Quotient Identities

$$\begin{array}{rlrlr}\tan (\theta )& ={\displaystyle \frac{\sin (\theta )}{\cos (\theta )}}& \cot (\theta )& ={\displaystyle \frac{\cos (\theta )}{\sin (\theta )}}& \end{array}$$

Pythagorean Identities

$$\begin{array}{rl}{\sin }^2(\theta )+{\cos }^2(\theta )& =1\\ {\tan }^2(\theta )+1& ={\sec }^2(\theta )\\ 1+{\cot }^2(\theta )& ={\csc }^2(\theta )\end{array}$$

Cofunction Identities

$$\begin{array}{rlrlrl}\sin (\theta )& =\cos (\frac{\pi }{2}-\theta )& \tan (\theta )& =\cot (\frac{\pi }{2}-\theta )& \sec (\theta )& =\csc (\frac{\pi }{2}-\theta )\\ \cos (\theta )& =\sin (\frac{\pi }{2}-\theta )& \cot (\theta )& =\tan (\frac{\pi }{2}-\theta )& \csc (\theta )& =\sec (\frac{\pi }{2}-\theta )\end{array}$$

Even-Odd Identities

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

Periodic Identities

$$\begin{array}{rlrl}\sin (\theta +2\pi )& =\sin (\theta )& \tan (\theta +\pi )& =\tan (\theta )\\ \csc (\theta +2\pi )& =\csc (\theta )& \cot (\theta +\pi )& =\cot (\theta )\\ \cos (\theta +2\pi )& =\cos (\theta )\\ \sec (\theta +2\pi )& =\sec (\theta )\end{array}$$

Sum and Difference Identities

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

Double Angle Identities

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

Half Angle Identities

$$\begin{array}{rl}|\sin \left({\displaystyle \frac{\theta }{2}}\right)|& =\sqrt{{\displaystyle \frac{1-\cos (\theta )}{2}}}\\ |\cos \left({\displaystyle \frac{\theta }{2}}\right)|& =\sqrt{{\displaystyle \frac{1+\cos (\theta )}{2}}}\\ \tan \left({\displaystyle \frac{\theta }{2}}\right)& ={\displaystyle \frac{1-\cos (\theta )}{\sin (\theta )}}\\ & ={\displaystyle \frac{\sin (\theta )}{1+\cos (\theta )}}\end{array}$$

This is all the identities we'll use in class. There are a few others that sometimes come in handy for other applications, so I'm including them below for the sake of completeness. It's a good exercise to try to prove them from the sum and difference identities, using systems of equations!

Product-to-Sum Identities

$$\begin{array}{rl}\sin (\alpha )\sin (\beta )& =\frac{1}{2}(\cos (\alpha -\beta )-\cos (\alpha +\beta ))\\ \cos (\alpha )\cos (\beta )& =\frac{1}{2}(\cos (\alpha -\beta )+\cos (\alpha +\beta ))\\ \sin (\alpha )\cos (\beta )& =\frac{1}{2}(\sin (\alpha +\beta )+\sin (\alpha -\beta ))\\ \cos (\alpha )\sin (\beta )& =\frac{1}{2}(\sin (\alpha +\beta )-\sin (\alpha -\beta ))\end{array}$$

Sum-to-Product Identities

$$\begin{array}{rl}\sin (\alpha )+\sin (\beta )& =2\sin \left({\displaystyle \frac{\alpha +\beta }{2}}\right)\cos \left({\displaystyle \frac{\alpha -\beta }{2}}\right)\\ \sin (\alpha )-\sin (\beta )& =2\cos \left({\displaystyle \frac{\alpha +\beta }{2}}\right)\sin \left({\displaystyle \frac{\alpha -\beta }{2}}\right)\\ \cos (\alpha )+\cos (\beta )& =2\cos \left({\displaystyle \frac{\alpha +\beta }{2}}\right)\cos \left({\displaystyle \frac{\alpha -\beta }{2}}\right)\\ \cos (\alpha )-\cos (\beta )& =-2\sin \left({\displaystyle \frac{\alpha +\beta }{2}}\right)\sin \left({\displaystyle \frac{\alpha -\beta }{2}}\right)\end{array}$$

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Posted by Bill Shillito