Chapter 14.4, Problem 54SR

To determine

To Prove:Given equation is identity in its domain.

Explanation of Solution

Given information:An equation ${\sin }^2\theta +{\tan }^2\theta =(1-{\cos }^2\theta )+\frac{{\sec }^2\theta }{co{\sec }^2\theta }$ is given.

Formula used: ${\sin }^2\theta =1-{\cos }^2\theta ,\tan \theta =\frac{\sin \theta }{\cos \theta },\sec \theta =\frac{1}{\cos \theta }and\sin \theta =\frac{1}{\cos ec\theta }$

Calculation:

  $\begin{array}{l}LHS={\sin }^2\theta +{\tan }^2\theta \\ =1-{\cos }^2\theta +\frac{{\sin }^2\theta }{{\cos }^2\theta }\\ =1-{\cos }^2\theta +\frac{1}{\cos e{c}^2\theta }\times {\sec }^2\theta \\ =1-{\cos }^2\theta +\frac{se{c}^2\theta }{\cos e{c}^2\theta }\\ =RHS\\ Hencegivenequationisanidentityinitsdomain.\end{array}$