5 Let $x_{1}, x_{2}, x_{3}, x_{4}$ all be positive real numbers, and $x_{1}+x_{2}+x_{3}+x_{4}=\pi$. Find the minimum value of 
$A=\left(2 \sin ^{2} x_{1}+\frac{1}{\sin ^{2} x_{1}}\right)\left(2 \sin ^{2} x_{2}+\frac{1}{\sin ^{2} x_{2}}\right)\left(2 \sin ^{2} x_{3}+\frac{1}{\sin ^{2} x_{3}}\right)\left(2 \sin ^{2} x_{4}+\right.$
$\left.\frac{1}{\sin ^{2} x_{4}}\right)$. (1991 Chinese National Training Team Test Question)
5. First note, if $x_{1}+x_{2}=a$ (a constant), then from $2 \sin x_{1} \sin x_{2}=\cos \left(x_{1}-x_{2}\right)-\cos a$, and $\left|x_{1}-x_{2}\right|\frac{\pi}{4}>x_{2}$, then fixing $x_{3}, x_{4}$, perform a smoothing transformation on $x_{1}, x_{2}$: that is, let $x_{1}^{\prime}=\frac{\pi}{4}$, $x_{2}^{\prime}=x_{1}+x_{2}-\frac{\pi}{4}, x_{3}^{\prime}=x_{3}, x_{4}^{\prime}=x_{4}$, then $x_{1}^{\prime}+x_{2}^{\prime}=x_{1}+x_{2},\left|x_{1}^{\prime}-x_{2}^{\prime}\right|g\left(2 \sin ^{2} x_{1}^{\prime} \sin ^{2} x_{2}^{\prime}\right)$

Therefore, $f\left(x_{1}, x_{2}\right)=2 g\left(2 \sin ^{2} x_{1} \sin ^{2} x_{2}\right)+2\left(\frac{\sin ^{2} x_{1}}{\sin ^{2} x_{2}}+\frac{\sin ^{2} x_{2}}{\sin ^{2} x_{1}}\right)$
$$\begin{array}{l}
\quad>2 g\left(2 \sin ^{2} x_{1}^{\prime} \sin ^{2} x_{2}^{\prime}\right)+2\left(\frac{\sin ^{4} x_{1}^{\prime}+\sin ^{4} x_{2}^{\prime}}{\sin ^{2} x_{1}^{\prime} \sin ^{2} x_{2}^{\prime}}\right)=f\left(x_{1}^{\prime}, x_{2}^{\prime}\right), \\
\quad A\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=f\left(x_{1}, x_{2}\right) f\left(x_{3}, x_{4}\right)>f\left(x_{1}^{\prime}, x_{2}^{\prime}\right) f\left(x_{3}^{\prime}, x_{4}^{\prime}\right)= \\
A\left(x_{1}^{\prime}, x_{2}^{\prime}, x_{3}^{\prime}, x_{4}^{\prime}\right)
\end{array}$$

Thus, we smooth $x_{1}$ to $\frac{\pi}{4}$, reducing the function value. If $x_{2}^{\prime}, x_{3}^{\prime}, x_{4}^{\prime}$ are still not equal to $\frac{\pi}{4}$, then continue the above transformation. It is evident that $A$ reaches its minimum when $x_{1}=x_{2}=x_{3}=x_{4}=\frac{\pi}{4}$.

In summary, the minimum value of $A$ is
$$\left(2 \sin ^{2} \frac{\pi}{4}+\frac{1}{\sin ^{2} \frac{\pi}{4}}\right)^{4}=81$$