Chapter 14.4, Problem 53SR

To determine

To Prove: $\cot \theta +\sec \theta =\frac{{\cos }^2\theta +\sin \theta }{\sin \theta \cos \theta }$ is identity.

Explanation of Solution

Given information:An equation is given $\cot \theta +\sec \theta =\frac{{\cos }^2\theta +\sin \theta }{\sin \theta \cos \theta }$

Formula used: $\cot \theta =\frac{\cos \theta }{\sin \theta }and\sec \theta =\frac{1}{\cos \theta }$

Calculation:

  $\begin{array}{l}LHS=\cot \theta +\sec \theta \\ =\frac{\cos \theta }{\sin \theta }+\frac{1}{\cos \theta }\\ =\frac{{\cos }^2\theta +\sin \theta }{\sin \theta \cos \theta }\\ =RHS\\ Hencegivenequationisanidentityinitsdomain.\end{array}$