Answer
See the proof.
Work Step by Step
\begin{align*}
\frac{d }{dx} \csc x&=\frac{d }{dx} \frac{1}{\sin x}\\
&=\frac{\sin x(0)-\cos x}{\sin^2 x}\\
&=\frac{-\cos x}{\sin^2 x}\\
&=-\frac{1}{\sin x}\frac{\cos x}{\sin x}\\
&=-\csc x \cot x.
\end{align*}
