Answer
$ \cot t $
Work Step by Step
Given expression is-
$ \frac{\sin t}{1 - \cos t} - \csc t $
= $ \frac{\sin t}{1 - \cos t} - \frac{1}{\sin t} $
= $\frac{ \sin^{2} t - (1 - \cos t)}{(1 - \cos t) \sin t} $
= $\frac{ (1 -\cos^{2} t) - (1 - \cos t)}{(1 - \cos t) \sin t} $
(using first Pythagorean identity)
= $\frac{ (1 -\cos t) (1 +\cos t) - (1 - \cos t)}{(1 - \cos t) \sin t} $
= $\frac{ (1 -\cos t) (1 +\cos t - 1 )}{(1 - \cos t) \sin t} $
= $\frac{ (1 -\cos t) \cos t }{(1 - \cos t) \sin t} $
= $\frac{ \cos t }{\sin t} $
= $ \cot t $
