Chapter 13.6, Problem 38WE

To determine

To Prove: The given identity.

Explanation of Solution

Given:

The given identity is ${\cos }^4\theta {\tan }^2\theta +{\sin }^2\theta {\tan }^2\theta +1={\sec }^2\theta$ .

Calculation:

Consider the given identity is,

  ${\cos }^4\theta {\tan }^2\theta +{\sin }^2\theta {\tan }^2\theta +1={\sec }^2\theta$

Solve the above equation as,

  $\begin{array}{l}{\cos }^4\theta {\tan }^2\theta +{\sin }^2\theta {\tan }^2\theta +1={\sec }^2\theta \\ {\tan }^2\theta \left({\cos }^4\theta +{\sin }^2\theta \right)+1={\sec }^2\theta \\ {\tan }^2\theta +1={\sec }^2\theta \\ {\sec }^2\theta ={\sec }^2\theta \end{array}$

Hence, the given identity is proved.