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