9. Find the minimum value of the function $(\sqrt{2} \sin x-3 \tan y)^{2}+(\sqrt{2} \cos x-3 \cot y)^{2}$, where $x, y \in\left(0, \frac{\pi}{2}\right)$.
9. Since $(\sqrt{2} \sin x-2 \tan y)^{2}+(\sqrt{2} \cos x-3 \cot y)^{2}$ can be regarded as the square of the distance between two points $A(\sqrt{2} \sin x, \sqrt{2} \cos x)$ and $B(3 \tan y, 3 \cot y)$ in the Cartesian coordinate system. At this time, the parametric equation of the trajectory of the moving point $A$ is $\left\{\begin{array}{l}x_{1}=\sqrt{2} \sin x \\ y_{1}=\sqrt{2} \cos x\end{array}\right.$

which is $x_{1}^{2}+y_{1}^{2}=2 \quad\left(x_{1}>0, y_{1}>0\right)$, its graph is an arc of a circle in the first quadrant; the parametric equation of the trajectory of the moving point $B$ is $\left\{\begin{array}{l}x_{2}=3 \tan y \\ y_{2}=3 \cot y\end{array}\right.$ which is $x_{2} y_{2}=9\left(x_{2}>0, y_{2}>0\right)$, its graph is a branch of a hyperbola in the first quadrant. From the graph, it can be seen that when and only when $x=y=\frac{\pi}{4}$, $A$ and $B$ have the shortest distance.
$$
\begin{aligned}
\text { Also }|O A| & =\sqrt{2}, \\
|O B| & =\sqrt{3^{2}+3^{2}}=3 \sqrt{2},
\end{aligned}
$$

then $|A B|=3 \sqrt{2}-\sqrt{2}=2 \sqrt{2}$,
thus the minimum value of the original function is $|A B|^{2}=(2 \sqrt{2})^{2}=8$.