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Answer 1 · 2017-03-02T19:01:33

see below

Use the following Properties:
1. Pythagorean Identity
$1 + {cot}^{2} x = {csc}^{2} x$ $1+cot^2x=csc^2x$ and solve for ${cot}^{2} x$ $cot^2x$

Left Hand Side:
$\frac{{cot}^{2} x}{csc x - 1} = \frac{{csc}^{2} x - 1}{csc x - 1}$ $cot^2x/(csc x-1)=(csc^2x-1)/(cscx-1)$

$= \frac{(csc x + 1) (csc x - 1)}{csc x - 1}$ $=((csc x+1)(cscx-1))/(cscx -1)$ ->factor the difference of 2 squares above

$=((csc x+1)cancel((cscx-1)))/cancel((cscx -1))$

$=cscx+1$

$=1/sinx +1$

$=1/sinx+sinx/sinx$

$=(1+sinx)/sinx$

$:.=$ Right Hand Side