#### Answer

\[\frac{{dy}}{{dz}} = {\cos ^2}z - {\sin ^2}z\]

#### Work Step by Step

\[\begin{gathered}
y = \frac{1}{{\sec z\csc z}} \hfill \\
\hfill \\
rewrite\,\,use\,trigonometric\,identities \hfill \\
\hfill \\
y = \cos z\sin z \hfill \\
\hfill \\
{\text{using}}\,\,the\,\,product\,rule. \hfill \\
\hfill \\
\frac{{dy}}{{dz}} = \cos z\,\left( {\cos z} \right) + \sin z\,\left( { - \sin z} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dz}} = {\cos ^2}z - {\sin ^2}z \hfill \\
\hfill \\
\end{gathered} \]