# How do you prove {cos(x)cot(x)}/{1-sin(x)} - 1 = csc(x)?

May 28, 2016

$L H S = \frac{\cos \left(x\right) \cot \left(x\right)}{1 - \sin \left(x\right)} - 1$

$= \frac{\cos \left(x\right) \cot \left(x\right)}{1 - \sin \left(x\right)} \sin \frac{x}{\sin} x - 1$

$= \frac{\frac{\cos x \cdot \cos x}{\cancel{\sin}} x \cdot \cancel{\sin} x}{\text{(1-sinx)sinx}} - 1$

$= {\cos}^{2} \frac{x}{\text{(1-sinx)sinx}} - 1$

$= \frac{1 - {\sin}^{2} x}{\text{(1-sinx)sinx}} - 1$

$= \frac{\left(1 + \sin x\right) \cancel{\left(1 - \sin x\right)}}{\cancel{\left(1 - \sin x\right)} \cdot \sin x} - 1$

$= \frac{\left(1 + \sin x\right) - \sin x}{\sin} x$
$= \frac{1 + \cancel{\sin} x - \cancel{\sin} x}{\sin} x = \frac{1}{\sin} x = \csc x = R H S$
proved