Question

the function y= x/tanx is ( an even, an odd, neither an even or odd) function, the function y= secx/x is (an even, an odd, neither an even or odd) function? I have to select one of the answers in parentheses.

Answer 1

Answer:

If $v(x)=\frac{x}{\tan(x)}$, then $v$ is even.

If $w(x)=\frac{\sec(x)}{x}$, then $w$ is odd.

Step-by-step explanation:

Summary of rules/ what we need:

$f(-x)=f(x)$ implies $f$ is even.

$f(-x)=-f(x)$ implies $f$ is odd.

So in either case, we need to replace $x$ with $-x$.

Let's begin.

First Problem:

$v(x)=\frac{x}{\tan(x)}$

Replace $x$ with $-x$:

$v(-x)=\frac{-x}{\tan(-x)}$

$v(-x)=\frac{-x}{-\tan(x)}$ (We used $\tan(x)$ is odd; that is, $\tan(-x)=-\tan(x)$)

$v(-x)=\frac{x}{\tan(x)}$

$v(-x)=v(x)$

This implies $v$ is an even function.

Second Problem:

$w(x)=\frac{\sec(x)}{x}$

Replace $x$ with $-x$:

$w(-x)=\frac{\sec(-x)}{-x}$

$w(-x)=\frac{\sec(x)}{-x}$ (We used $\sec(x)$ is even; that is, $\sec(-x)=\sec(x)$)

$w(-x)=-\frac{\sec(x)}{x}$

$w(-x)=-w(x)$

This implies $w$ is an odd function.