Chapter 5.2, Problem 92E

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:

  $1+\tan x=\sec x$

The given equation is not an identity because it is not true for all values of $x$ . For example:

Let $x=\pi$

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

  $\begin{array}{l}\text{ 1+}\tan x\\ =1+\tan \pi \text{ [Substitute value of }x\text{]}\\ =1+0\text{ [Put value of tan}\pi ]\\ =1\text{ [Add]}\end{array}$

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

  $\begin{array}{l}\sec x\\ =\sec \pi \text{ [Substitute value of }x\text{]}\\ =-1\text{ [Put value of sec}\pi \text{]}\end{array}$

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

Hence for $x=\pi$ , 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 ${\text{Y}}_1=\text{1+}\tan x$ and ${\text{Y}}_2=\sec x$ 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.