Answer
\[ = - 1\]
Work Step by Step
\[\begin{gathered}
\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\,\,\,\frac{{\cos x}}{{x - \frac{\pi }{2}}}\,\, \hfill \\
\hfill \\
let\,,\,\,\,\cos x = \sin \,\left( {\frac{\pi }{2} - 2} \right) \hfill \\
\hfill \\
= - \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\sin \,\left( {\frac{\pi }{2} - x} \right)}}{{\frac{\pi }{2} - x}} \hfill \\
\hfill \\
using\,\,special\,\,\,limit:\,\,\,\mathop {\lim }\limits_{x \to 0} \,\,\frac{{\sin x}}{x} = 1 \hfill \\
then \hfill \\
\hfill \\
- \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\sin \,\left( {\frac{\pi }{2} - x} \right)}}{{\frac{\pi }{2} - x}} = - 1 \hfill \\
\end{gathered} \]
