Question

Determine whether the function is even, odd, or neither.

$f\left(x\right)=\cos \left(\sin x\right)$

Short answer

The$f\left(x\right)=\cos \left(\sin x\right)$isan even function.

Step-by-step solution

Step 1. State the definitions of even and odd functions.

A function $f\left(x\right)$is an even function if $f\left(x\right)=f\left(-x\right)$.

A function $f\left(x\right)$ is an odd function if $f\left(x\right)=f\left(-x\right)$.

Step 2. Calculate f−x. 

Substitute $-x$ for x into the function $f\left(x\right)=\cos \left(\sin x\right)$and simplify.

$$\begin{gathered} f\left(x\right)=\cos \left(\sin x\right) \\ f\left(-x\right)=\cos \left(\sin \left(-x\right)\right) \end{gathered}$$

Step 3. Simplify.

Note: $\sin \left(-t\right)=-\sin t$, that is., sine function, in general, is an odd function and $\cos \left(-t\right)=\cos t$, that is., cosine function, in general, is an even function.

$f\left(-x\right)=\cos \left(\sin \left(-x\right)\right)$

$$\begin{gathered} =\cos \left(-\sin x\right) \\ =\cos \left(\sin x\right) \\ =f\left(x\right) \end{gathered}$$

Since, $f\left(-x\right)=f\left(x\right)$, therefore, $f\left(x\right)=\cos \left(\sin x\right)$ is an even function.