Answer
$\dfrac{d}{dx}( \csc x)= - \csc x \cot x$
Work Step by Step
Need to prove $\dfrac{d}{dx}( \csc x)= - \csc x \cot x$
Consider left hand side.
$\dfrac{d}{dx}( \csc x)=\dfrac{d}{dx}( \dfrac{1}{\sin x})\\=\dfrac{- \cos x}{ \sin^2 x}\\=(\dfrac{-1}{\sin x})(\dfrac{\cos x}{\sin x})\\= - \csc x \cot x$
