Answer
See the proof below.
Work Step by Step
\begin{align*}
\frac{d }{dx} \cot x&=\frac{d }{dx} \frac{\cos x}{\sin x}\\
&=\frac{\sin x(-\sin x)-\cos x(\cos x)}{\sin^2 x}\\
&=\frac{-\sin^2 x-\cos^2 x}{\sin^2 x}\\
&=\frac{-1}{\sin^2 x}\\
&=-\csc^2 x.
\end{align*}
