$\mathop {\lim }\limits_{x \to 0}\frac{\cos \left ( \frac{\pi }{2}\cos x \right )}{\sin \left ( \tan x \right )}=\mathop {\lim }\limits_{x \to 0}\frac{\sin \left ( \frac{\pi }{2}-\frac{\pi }{2}\cos x \right )}{\sin \left ( \tan x \right )}=\mathop {\lim }\limits_{x \to 0}\frac{\sin \left ( \pi\sin^2 \frac{x}2 \right )}{\sin \left ( \tan x \right )}$
$=\mathop {\lim }\limits_{x \to 0}\pi.\frac{\sin \left ( \pi\sin^2 \frac{x}2 \right )}{ \pi\sin^2 \frac{x}2}.\frac{\sin^2 \frac{x}2}{\frac{x^2}4}.\frac14.\frac{\tan x}{\sin \left ( \tan x \right )}.\frac{x}{\tan x}.x =\pi.1.1.\frac14.1.1.0=0$