Chapter 5.2, Problem 89E

To determine

To explain: Why the given equation is not an identity and find one value of the variable for which the equation is not true.

Explanation of Solution

Given information:

The following equation:

  $\sqrt{{\sec }^{2}x-1}=\tan x$

Formula used:

The following identity:

  ${\text{1+tan}}^{2}x={\sec }^{2}x$

The equation can be simplified as:

  $\begin{array}{l}\sqrt{{\sec }^{2}x-1}=\tan x\\ \sqrt{{\tan }^{2}x}=\tan x\text{ [Pythagorean identity]}\\ \pm \tan x\ne \tan x\text{ [Square root]}\end{array}$

The left hand side is not equal to the right hand side. Hence the equation is not an identity.

Let $x=\frac{2\pi }{3}$

Substituting value of $x$ on left hand side of the equation,

  $\begin{array}{l}\sqrt{{\sec }^{2}x-1}\\ =\sqrt{{\sec }^2\frac{2\pi }{3}-1}\text{ [Substitute value of }x]\\ =\sqrt{{(-2)}^2-1}\text{ [Put value of sec }\frac{2\pi }{3}]\\ =\sqrt{4-1}\text{ [Square]}\\ =\sqrt{3}\text{ [Subtract]}\end{array}$

Substituting value of $x$ on right hand side of the equation,

  $\begin{array}{l}\tan x\\ =\tan \frac{2\pi }{3}\text{ [Substitute value of }x]\\ =-\sqrt{3}\text{ [Put value of }\tan \frac{2\pi }{3}]\end{array}$

The value on the left hand side is not equal to the value on the right hand side.

Hence for $x=\frac{2\pi }{3}$ , the equation is not true

Also, the given equation is not an identity because the expressions on the left hand side and the right hand side when plotted on a graph, do not coincide.

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y}=}$ button.

Step 3: Enter the expressions ${\text{Y}}_1=\sqrt{{\sec }^{2}x-1}$ and ${\text{Y}}_2=\tan x$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

The graphs of the left hand side expression and the right hand side expression do no coincide.