Proof: $\because \sin ^{2} \alpha+\cos ^{2} \alpha=1$, and the function $f(x)=$ $x^{5}$ is convex on $[0,1]$, hence by Jensen's inequality we have
$$\begin{aligned} {\left[\sin ^{2} \alpha\right]^{5}+\left[\cos ^{2} \alpha\right]^{5} } & \geqslant 2 \cdot\left[\frac{\sin ^{2} \alpha+\cos ^{2} \alpha}{2}\right]^{5} \\ & =2 \cdot \frac{1}{2^{5}}=\frac{1}{16} . \end{aligned}$$