15. Given that $\alpha$ is an acute angle, find the minimum value of $\left(1+\frac{1}{\sin \alpha}\right)\left(1+\frac{1}{\cos \alpha}\right)$.
15. $\left(1+\frac{1}{\sin \alpha}\right)\left(1+\frac{1}{\cos \alpha}\right) \geqslant\left(1+\frac{1}{\sqrt{\sin \alpha \cos \alpha}}\right)^{2}=\left[1+\frac{\sqrt{2}}{\sqrt{\sin 2 \alpha}}\right]^{2} \geqslant(1+\sqrt{2})^{2}=$ $3+2 \sqrt{2}$, when and only when $\sin \alpha=\cos \alpha$, and $\sin 2 \alpha=1$, the minimum value is obtained, at this time $\alpha=\frac{\pi}{4}$. Therefore, the minimum value of $\left(1+\frac{1}{\sin \alpha}\right)\left(1+\frac{1}{\cos \alpha}\right)$ is $3+2 \sqrt{2}$.