Example 10 Find the minimum value of the function $f(x, y)=\frac{3}{2 \cos ^{2} x \sin ^{2} x \cos ^{2} y}+$ $\frac{5}{4 \sin ^{2} y}\left(x, y \neq \frac{k \pi}{2}, k \in Z\right)$. (Example 3 in [4])
$$\begin{aligned}
& \text { Solve } \frac{3}{2 \cos ^{2} x \sin ^{2} x \cos ^{2} y}+\frac{5}{4 \sin ^{2} y} \\
= & \frac{3\left(\sin ^{2} x+\cos ^{2} x\right)}{2 \cos ^{2} x \sin ^{2} x \cos ^{2} y}+\frac{5}{4 \sin ^{2} y} \\
= & \frac{3}{2 \cos ^{2} x \cos ^{2} y}+\frac{3}{2 \sin ^{2} x \cos ^{2} y}+\frac{5}{4 \sin ^{2} y} \\
= & \frac{6}{4 \cos ^{2} x \cos ^{2} y}+\frac{6}{4 \sin ^{2} x \cos ^{2} y}+\frac{5}{4 \sin ^{2} y} \\
= & \frac{(\sqrt{6})^{2}}{4 \cos ^{2} x \cos ^{2} y}+\frac{(\sqrt{6})^{2}}{4 \sin ^{2} x \cos ^{2} y}+\frac{(\sqrt{5})^{2}}{4 \sin ^{2} y} \\
\geqslant & \frac{(\sqrt{6}+\sqrt{6}+\sqrt{5})^{2}}{4 \cos ^{2} x \cos ^{2} y+4 \sin ^{2} x \cos ^{2} y+4 \sin ^{2} y} \\
= & \frac{29+4 \sqrt{30}}{4} .
\end{aligned}$$

When and only when $\frac{\sqrt{6}}{4 \cos ^{2} x \cos ^{2} y}=\frac{\sqrt{6}}{4 \sin ^{2} x \cos ^{2} y}=\frac{\sqrt{5}}{4 \sin ^{2} y}$,
i.e., $\tan ^{2} x=1, \tan ^{2} y=\frac{\sqrt{30}}{12}$, the equality holds, thus the minimum value is $\frac{29+4 \sqrt{30}}{4}$.