Answer
As left side transforms into right side, hence given identity-
$\frac{\csc\theta - 1}{\cot\theta} $ = $ \frac{\cot\theta}{\csc\theta + 1}$ is true.
Work Step by Step
Given identity is-
$\frac{\csc\theta - 1}{\cot\theta} $ = $ \frac{\cot\theta}{\csc\theta + 1}$
Taking L.S.
$\frac{\csc\theta - 1}{\cot\theta} $
= $\frac{\csc\theta - 1}{\cot\theta} . \frac{\csc\theta + 1}{\csc\theta + 1} $
{Multiplying the numerator and denominator by , $(\csc\theta + 1)$}
= $\frac{(\csc\theta - 1)(\csc\theta + 1)}{\cot\theta (\csc\theta + 1)} $
= $\frac{ \csc^{2} \theta - 1^{2}} {\cot\theta (\csc\theta + 1)}$
{Recall $(a-b)(a+b) $ = $a^{2} - b^{2}$ }
= $\frac{ \cot^{2} \theta} {\cot\theta (\sec\theta - 1)}$
( From third Pythagorean identity, $\csc^{2}\theta - 1$ = $\cot^{2}\theta$)
=$ \frac{\cot\theta}{\csc\theta + 1}$
= R.S.
As left side transforms into right side, hence given identity-
$\frac{\csc\theta - 1}{\cot\theta} $ = $ \frac{\cot\theta}{\csc\theta + 1}$ is true.
