Answer
As left side transforms into right side, hence given identity-
$ (\cot x - \csc x)(\cos x + 1)$ = $ - \sin x$ is verified as true.
Work Step by Step
Given identity is-
$ (\cot x - \csc x)(\cos x + 1)$ = $ - \sin x$
Taking L.S.
$ (\cot x - \csc x)(\cos x + 1)$
= $ [\frac{\cos x }{\sin x} - \frac{1}{\sin x}] (\cos x + 1)$
= $ (\frac{\cos x -1 }{\sin x}) (\cos x + 1)$
= $ \frac{(\cos x -1) (\cos x + 1) }{\sin x} $
= $ \frac{\cos^{2} x -1 }{\sin x} $
{Recall $(a-b)(a+b) $ = $a^{2} - b^{2}$ }
= $ - \frac{1 - \cos^{2} x }{\sin x} $
= $ - \frac{\sin^{2} x }{\sin x} $ (Recall first Pythagorean identity)
= $ - \sin x$
= R.S.
As left side transforms into right side, hence given identity-
$ (\cot x - \csc x)(\cos x + 1)$ = $ - \sin x$ is verified as true.
