3. Among the 4 functions $y=\sin |x|, y=\cos |x|, y=|\cot x|, y=\lg |\sin x|$, the even function that is periodic with $\pi$ and monotonically increasing in $\left(0, \frac{\pi}{2}\right)$ is ().
A. $y=\sin |x|$
B. $y=\cos |x|$
C. $y=|\cot x|$
D. $y=\lg |\sin x|$
3. D $y=\sin |x|$ is not a periodic function. $y=\cos |x|=\cos x$ has a period of $2 \pi$. $y=|\cot x|$ is monotonically decreasing on $\left(0, \frac{\pi}{2}\right)$. Only $y=\lg |\sin x|$ satisfies all the conditions.