Example 7. Let $s_{1}, x_{2}, c_{3}, x_{4}$ all be positive real numbers, and $x_{1} + x_{2} + x_{3} - x_{4} = \pi$. Find the minimum value of the expression
\[
\left(2 \sin ^{2} x_{1}+\frac{1}{\sin ^{2} x_{1}}\right)\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right)
- \left(2 \sin ^{2} x_{3}+\frac{1}{\sin ^{2} x_{3}}\right)\left(2 \sin ^{2} x_{4}+\frac{1}{\sin ^{2} x_{4}}\right).
\]
(1991, China National Training Team Test Question)
Let $x_{1}+x_{2}$ be a constant. Since
$$
\begin{array}{l}
\sin x_{1} \sin x_{2} \\
=\frac{1}{2}\left[\cos \left(x_{1}-x_{2}\right)-\cos \left(x_{1}+x_{2}\right)\right],
\end{array}
$$

we know that the value of $\sin x_{1} \sin x_{2}$ increases as $\left|x_{1}-x_{2}\right|$ decreases. Let the expression in question be $f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$. If $x_{1}, x_{2}, x_{3}, x_{4}$ are not all equal, without loss of generality, assume $x_{1}>\frac{\pi}{4}>x_{2}$. Let $x_{1}{ }^{\prime}=\frac{\pi}{4}, x_{2}{ }^{\prime}=x_{1}+x_{2}-\frac{\pi}{4}$,
$$
x_{3}{ }^{\prime}=x_{3}, x_{4}{ }^{\prime}=x_{4},
$$

Thus, we have $x_{1}{ }^{\prime}+x_{2}{ }^{\prime}=x_{1}+x_{2}$,
$\left|x_{1}{ }^{\prime}-x_{2}{ }^{\prime}\right|2\left(2 \sin ^{2} x_{1}^{\prime} \sin ^{2} x_{2}{ }^{\prime}+\frac{1}{2 \sin ^{2} x_{1} \sin ^{2} x_{2}{ }^{\prime}}\right) \\
\quad+2\left(\frac{\sin ^{2} x_{1}^{\prime}}{\sin ^{2} x_{2}{ }^{\prime}}+\frac{\sin ^{2} x_{2}^{\prime}}{\sin ^{2} x_{1}{ }^{\prime}}\right) \\
=\left(2 \sin ^{2} x_{1}^{\prime}+\frac{1}{\sin ^{2} x_{1}{ }^{\prime}}\right) \\
\quad \cdot\left(2 \sin ^{2} x_{2}{ }^{\prime}+\frac{1}{\sin ^{\prime} x_{2}^{\prime}}\right) .
\end{array}
$$

Thus,
$$
f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)>f\left(x_{1}{ }^{\prime}, x_{2}{ }^{\prime}, x_{3}{ }^{\prime}, x_{4}{ }^{\prime}\right) .
$$

If $x_{2}{ }^{\prime}, x_{3}{ }^{\prime}, x_{4}{ }^{\prime}$ are not all equal, then we can perform a similar smoothing transformation to prove that when $x_{1}, x_{2}, x_{3}, x_{4}$ are not all equal, we always have
$$
f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)>f\left(\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}\right) .
$$

Therefore, the minimum value is $f\left(\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}\right)$
$=81$, which is achieved when and only when $x_{1}=x_{2}=x_{3}=x_{4}=\frac{\pi}{4}$.