#### Answer

\[\frac{{dy}}{{d\theta }} = - 2{\csc ^3}\theta \,\cot \,\theta \]

#### Work Step by Step

\[\begin{gathered}
y = {\csc ^2}\theta - 1 \hfill \\
\hfill \\
differentiate\,using\,\,the\,chain\,\,rule: \hfill \\
\hfill \\
\frac{{dy}}{{d\theta }} = 2{\csc ^2}\theta \,{\left( {\csc \,\,\theta } \right)^\prime } \hfill \\
\hfill \\
\frac{{dy}}{{d\theta }} = 2{\csc ^2}\theta \,\left( { - \csc \,\theta \,\cot \theta } \right) \hfill \\
\hfill \\
multiply \hfill \\
\hfill \\
\frac{{dy}}{{d\theta }} = - 2{\csc ^3}\theta \,\cot \,\theta \hfill \\
\end{gathered} \]