## Precalculus (6th Edition) Blitzer

If $f\left( -x \right)=f\left( x \right)$ , then the function is even; if the function has symmetry about the y-axis, then it will be an even function.
We know that even trigonometric functions can be checked by the following steps: Step I: If $f\left( -x \right)=f\left( x \right)$ , then it will be an even function. Step II: Graphically, if the function has symmetry about the y-axis, then it will be an even function. \begin{align} & \sin \left( -x \right)=-\sin x \\ & \cos \left( -x \right)=\cos x \\ & \tan \left( -x \right)=-\tan x \\ \end{align} Similarly, \begin{align} & \operatorname{cosec}\left( -x \right)=-\operatorname{cosec}x \\ & \sec \left( -x \right)=\sec x \\ & \cot \left( -x \right)=-\cot x \\ \end{align} Thus, if any trigonometric function has $f\left( -x \right)=f\left( x \right)$, then it will be an even function, and among six trigonometric functions, $\cos x,\text{ and }\sec x$ are even trigonometric functions. For example: \begin{align} & \cos \left( -\frac{\pi }{4} \right)=\cos \frac{\pi }{4} \\ & \sec \left( -\frac{\pi }{4} \right)=\sec \frac{\pi }{4} \\ \end{align}