# The Cofunction and Odd-Even Identities

### The Cofunction Identities

$\begin{array}{ccc}\mathrm{sin}\left(\frac{\pi }{2}-x\right)=\mathrm{cos}\left(x\right)& \mathrm{cos}\left(\frac{\pi }{2}-x\right)=\mathrm{sin}\left(x\right)& \mathrm{tan}\left(\frac{\pi }{2}-x\right)=\mathrm{cot}\left(x\right)\\ \mathrm{csc}\left(\frac{\pi }{2}-x\right)=\mathrm{sec}\left(x\right)& \mathrm{sec}\left(\frac{\pi }{2}-x\right)=\mathrm{csc}\left(x\right)& \mathrm{cot}\left(\frac{\pi }{2}-x\right)=\mathrm{tan}\left(x\right)\end{array}$

Example:

Find the value of $\mathrm{cot}\left(60°\right)$ .

Use the co-function identity $\mathrm{tan}\left(90°-\theta \right)=\mathrm{cot}\left(\theta \right)$ to rewrite the problem.

$\begin{array}{l}\mathrm{cot}\left(60°\right)=\mathrm{tan}\left(90°-60°\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{tan}\left(30°\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{\sqrt{3}}{3}\end{array}$

### The Odd-Even Identities

$\mathrm{cos}\left(x\right)$ is an even function, $\mathrm{sin}\left(x\right)$ is an odd function as trigonometric functions for real variables.

$\begin{array}{ccc}\mathrm{sin}\left(-x\right)=-\mathrm{sin}\left(x\right)& \mathrm{cos}\left(-x\right)=\mathrm{cos}\left(x\right)& \mathrm{tan}\left(-x\right)=-\mathrm{tan}\left(x\right)\\ \mathrm{csc}\left(-x\right)=-\mathrm{csc}\left(x\right)& \mathrm{sec}\left(-x\right)=-\mathrm{sec}\left(x\right)& \mathrm{cot}\left(-x\right)=-\mathrm{cot}\left(x\right)\end{array}$

Example:

If $\mathrm{cos}\left(\alpha \right)=\frac{1}{2}$ , determine $\mathrm{cos}\left(-\alpha \right)$ and $\mathrm{sec}\left(-\alpha \right)$ .

$\begin{array}{l}\mathrm{cos}\left(-\alpha \right)=\mathrm{cos}\left(\alpha \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{2}\end{array}$

$\begin{array}{l}\mathrm{sec}\left(\alpha \right)=\frac{1}{\mathrm{cos}\left(\alpha \right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{1}{\left(\frac{1}{2}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2\end{array}$

$\begin{array}{l}\mathrm{sec}\left(-\alpha \right)=\mathrm{sec}\left(\alpha \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2\end{array}$