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