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The Cofunction and Odd-Even Identities

The Cofunction Identities

$\begin{matrix} \sin (\frac{π}{2} - x) = \cos (x) & \cos (\frac{π}{2} - x) = \sin (x) & \tan (\frac{π}{2} - x) = \cot (x) \\ \csc (\frac{π}{2} - x) = \sec (x) & \sec (\frac{π}{2} - x) = \csc (x) & \cot (\frac{π}{2} - x) = \tan (x) \end{matrix}$

Example:

Find the value of $\cot (60 °)$ .

Use the co-function identity $\tan (90 ° - θ) = \cot (θ)$ to rewrite the problem.

$\begin{array}{l} \cot (60 °) = \tan (90 ° - 60 °) \\ = \tan (30 °) \\ = \frac{\sqrt{3}}{3} \end{array}$

The Odd-Even Identities

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

$\begin{matrix} \sin (- x) = - \sin (x) & \cos (- x) = \cos (x) & \tan (- x) = - \tan (x) \\ \csc (- x) = - \csc (x) & \sec (- x) = - \sec (x) & \cot (- x) = - \cot (x) \end{matrix}$

Example:

If $\cos (α) = \frac{1}{2}$ , determine $\cos (- α)$ and $\sec (- α)$ .

$\begin{array}{l} \cos (- α) = \cos (α) \\ = \frac{1}{2} \end{array}$

$\begin{array}{l} \sec (α) = \frac{1}{\cos (α)} \\ = \frac{1}{(\frac{1}{2})} \\ = 2 \end{array}$

$\begin{array}{l} \sec (- α) = \sec (α) \\ = 2 \end{array}$