Answer
Please see below.
Work Step by Step
We simplify as follows:
$$\frac{\sin (x+ \pi )}{\cos (x+ \frac{3\pi }{2})}=\frac{\sin x \cos \pi + \cos x \sin \pi }{\cos x \cos \frac{3\pi }{2} - \sin x \sin \frac{3 \pi }{2}}=\frac{-\sin x }{ \sin x }=-1 = \tan ^2x-\sec ^2x$$(Please remember that $1+ \tan^2 x = \sec ^2 x$).
