5. Let $\alpha \in\left(0, \frac{\pi}{2}\right)$, then the minimum value of $\frac{\sin ^{3} \alpha}{\cos \alpha}+\frac{\cos ^{3} \alpha}{\sin \alpha}$ is
5. 1 .

Let $t=\sin \alpha \cos \alpha$, then $t=\frac{1}{2} \sin 2 \alpha \in\left(0, \frac{1}{2}\right]$, so
$$
\frac{\sin ^{3} \alpha}{\cos \alpha}+\frac{\cos ^{3} \alpha}{\sin \alpha}=\frac{\sin ^{4} \alpha+\cos ^{4} \alpha}{\sin \alpha \cos \alpha}=\frac{1-2(\sin \alpha \cos \alpha)^{2}}{\sin \alpha \cos \alpha}=\frac{1-2 t^{2}}{t} .
$$

Clearly, the function $f(t)=\frac{1-2 t^{2}}{t}=\frac{1}{t}-2 t$ is monotonically decreasing on $\left(0, \frac{1}{2}\right]$, hence the minimum value of $f(t)$ on $\left(0, \frac{1}{2}\right]$ is $f\left(\frac{1}{2}\right)=1$, i.e., the minimum value of $\frac{\sin ^{3} \alpha}{\cos \alpha}+\frac{\cos ^{3} \alpha}{\sin \alpha}$ is 1.