# How do you prove the following?

## $\cot x + \cot y = \frac{\csc x \csc y}{\csc} \left(x + y\right)$

$L H S = \cot x + \cot y$
$= \cos \frac{x}{\sin} x + \cos \frac{y}{\sin} y$
$= \frac{\sin x \cos y + \cos x \sin y}{\sin x \sin y}$
$= \sin \frac{x + y}{\sin x \sin y}$
$= \frac{\frac{1}{\csc} \left(x + y\right)}{\frac{1}{\csc x \csc y}} = \frac{\csc x \csc y}{\csc \left(x + y\right)} = R H S$