Chapter 4.5, Problem 62E

Solution

To determine: The value of $\cot x$ as a reflection about the $x$ -axis of a horizontal translation of $\tan x$ .

The value of $\cot x$ as a reflection about the $x$ -axis of a horizontal translation of $\tan x$ is $\cot x=-\tan \left(x-\frac{\pi }{2}\right)$ .

Given information:

The expression $\cot x$ .

Explanation:

Consider the expression $\cot x$ ,

The function $y=\cot x$ is a inverse of $y=\tan x$ .

A graph that is translated horizontally $h$ units to the right are of the form expression $g\left(x\right)=f\left(x\right)-h$ .

As the graph $y=\cot x$ is $\frac{\pi }{2}$ to the right of the graph $y=-\tan x$ , it is needed to translate the secant function horizontally by $\frac{\pi }{2}$ units to the right.

Take $h=\frac{\pi }{2}$ into $g\left(x\right)=f\left(x\right)-h$ .

  $\cot x=-\tan \left(x-\frac{\pi }{2}\right)$

Thus, the value of $\cot x$ as a reflection about the $x$ -axis of a horizontal translation of $\tan x$ is $\cot x=-\tan \left(x-\frac{\pi }{2}\right)$ .