Chapter 13.3, Problem 11MCQ

To determine

To verify: The given identity.

Explanation of Solution

Given:

The identify is given as:

  ${\cot }^2\theta +1=\frac{\cot \theta }{\cos \theta .\sin \theta }$

From identify:

  $\cot \theta =\frac{\cos \theta }{\sin \theta }$

Taking the left hand side of the identify:

  $\begin{array}{l}{\cot }^2\theta +1=\frac{{\cos }^2\theta +1}{{\sin }^2\theta }\\ =\frac{{\cos }^2\theta +{\sin }^2\theta }{{\sin }^2\theta }\end{array}$

From identify:

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

Therefore,

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

From identify:

  $\cot \theta =\frac{\cos \theta }{\sin \theta }$

Hence,

  $\begin{array}{l}{\cot }^2\theta +1=\frac{\cot \theta }{\sin \theta \cos \theta }\\ =\text{Right}\text{Hand}\text{Side}\end{array}$

Hence, from the above equation, the above identify is verified.