Chapter 5.2, Problem 69E

Solution

To find: A simple function for the given expression and prove that both are the same function.

The simple expression for the given expression is 1.

Given information:

The given expression is $\left({\sec }^{2}x\right)\left(1-{\sin }^{2}x\right)$ .

Formula used:

The trigonometric identities are $\sec x=\frac{1}{\cos x}$ and $1-{\sin }^{2}x={\cos }^{2}x$ .

Calculation:

The simple function of the given expression that has the same graph can be determined using the function’s graph.

Draw the graph of the expression $\left({\sec }^{2}x\right)\left(1-{\sin }^{2}x\right)$ . The graph of the function $\left({\sec }^{2}x\right)\left(1-{\sin }^{2}x\right)$ is shown below.

  

Figure (1)

From the obtained graph, it can be observed that the graph of given expression is the same as the graph of $y=1$ .

Therefore, the simple expression for the given expression is 1.

Simplify the given expression using the trigonometric identities, $\sec x=\frac{1}{\cos x}$ and $1-{\sin }^{2}x={\cos }^{2}x$ .

  $\begin{array}{c}\left({\sec }^{2}x\right)\left(1-{\sin }^{2}x\right)=\frac{1}{\left({\cos }^{2}x\right)}\left({\cos }^{2}x\right)\\ =1\end{array}$

It can be verified that the simple function for the given expression is 1.