Answer
$$\frac{1}{2}$$
Work Step by Step
We have
$$
\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\sin 2 x}=\frac{0}{0}
$$
is an intermediate form, then we can apply L’Hôpital’s Rule as follows
$$
\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\sin 2 x}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{-\sin x}{2\cos 2 x}=\frac{1}{2}.
$$
