Chapter 13.6, Problem 36WE

To determine

To Prove: The given identity.

Explanation of Solution

Given:

The given identity is ${\sec }^2\alpha +{\csc }^2\alpha ={\sec }^2\alpha {\csc }^2\alpha$ .

Calculation:

Consider the given identity is,

  ${\sec }^2\alpha +{\csc }^2\alpha ={\sec }^2\alpha {\csc }^2\alpha$

Solve the above equation by Pythagorean identity as,

  $\begin{array}{c}{\sec }^2\alpha +{\csc }^2\alpha ={\sec }^2\alpha \left(1+{\cot }^2\alpha \right)\\ ={\sec }^2\alpha +{\cot }^2\alpha {\sec }^2\alpha \\ ={\sec }^2\alpha +\frac{{\cos }^2\alpha }{{\cos }^2\alpha {\sin }^2\alpha }\\ ={\sec }^2\alpha +\frac{1}{{\sin }^2\alpha }\end{array}$

Solve further as,

  ${\sec }^2\alpha +{\csc }^2\alpha ={\sec }^2\alpha +{\csc }^2\alpha$

Hence, the given identity is proved.