Example 8 Find the minimum value of the function $y=\frac{5}{2 \sin ^{2} x+3}+\frac{8}{5 \cos ^{2} x+6}$. (Example 2 in [4])
Solve $y=\frac{5}{2 \sin ^{2} x+3}+\frac{8}{5 \cos ^{2} x+6}=\frac{5^{2}}{10 \sin ^{2} x+15}$ $+\frac{4^{2}}{10 \cos ^{2} x+12} \geqslant \frac{(5+4)^{2}}{10 \sin ^{2} x+15+10 \cos ^{2} x+12}=\frac{81}{37}$ when and only when $\frac{5}{10 \sin ^{2} x+15}=\frac{4}{10 \cos ^{2} x+12}$, i.e., $\sin ^{2} x=$ $\frac{5}{9}$, i.e., $\sin x= \pm \frac{\sqrt{5}}{5}$, the equality holds, thus the minimum value sought is $\frac{81}{37}$.