Chapter 13.3, Problem 10CYU

To determine

To Prove: The given expression is the identity.

Explanation of Solution

Given:

The given expression is $\tan \left(\theta +\frac{\pi }{2}\right)=-\cot \theta$ .

Calculation:

Consider the expression is,

  $\tan \left(\theta +\frac{\pi }{2}\right)=-\cot \theta$

Then, the sum of the identity is,

  $\begin{array}{c}\tan \left(\theta +\frac{\pi }{2}\right)=\frac{\tan \theta +\tan \frac{\pi }{2}}{1-\tan \theta \tan \frac{\pi }{2}}\\ =\frac{\frac{\tan \theta }{\tan \frac{\pi }{2}}+1}{\frac{1}{\tan \frac{\pi }{2}}-\tan \theta }\\ =-\frac{1}{\tan \theta }\end{array}$

Then, by the reciprocal identity,

  $\cot \theta =\frac{1}{\tan \theta }$

Then,

  $\begin{array}{c}\tan \left(\theta +\frac{\pi }{2}\right)=-\frac{1}{\tan \theta }\\ =-\cot \theta \end{array}$

Thus, the required identity is verified.