13. Let $x \geqslant y \geqslant z \geqslant \frac{\pi}{12}$, and $x+y+z=\frac{\pi}{2}$. Find the maximum and minimum values of the product $\cos x \sin y \cos z$.
$$
\begin{array}{l}
\cos x \sin y \cos z=\frac{1}{2} \cos x[\sin (y+z)+\sin (y-z)]=\frac{1}{2} \cos x[\cos x+\sin (y-z)] \\
\geqslant \frac{1}{2} \cos ^{2} x \geqslant \frac{1}{2} \cos ^{2} \frac{\pi}{3}=\frac{1}{8}, \text { equality holds when } x=\frac{\pi}{3}, y=z=\frac{\pi}{12} ; \\
\cos x \sin y \cos z=\frac{1}{2} \cos z[\sin (x+y)-\sin (x-y)]=\frac{1}{2} \cos z[\cos z-\sin (x-y)] \\
\leqslant \frac{1}{2} \cos ^{2} z \leqslant \frac{1}{2} \cos ^{2} \frac{\pi}{12}=\frac{2+\sqrt{3}}{8}, \text { equality holds when } z=\frac{\pi}{12}, x=y=\frac{5 \pi}{24} .
\end{array}
$$

Therefore, the maximum value of $\cos x \sin y \cos z$ is $\frac{2+\sqrt{3}}{8}$, and the minimum value is $\frac{1}{8}$.