Chapter 13.2, Problem 9PPS

To determine

To Verify : The expression $\cot \theta (\cot \theta +\tan \theta )={\csc }^2\theta$

Explanation of Solution

Given information :

We know that, the given value is

  $\begin{array}{l}\cot \theta (\cot \theta +\tan \theta )(L.H.S.)\\ {\mathrm{cosec}}^2\theta (R.H.S.)\end{array}$

Formula Used:

  ${\mathrm{cosec}}^2\theta -{\cot }^2\theta =1$

Proof :

The given equation is

L.H.S.

  $\begin{array}{l}\cot \theta (\cot \theta +\tan \theta )\\ ={\cot }^2\theta +\cot \theta \cdot \tan \theta \\ ={\cot }^2\theta +1\left[\because \tan \theta =\frac{1}{\cot \theta }\right]\\ ={\mathrm{cosec}}^2\theta =R.H.S.\end{array}$