Example 4 (2008 National I) If the line $\frac{x}{a}+\frac{y}{b}=1$ passes through the point $M(\cos \alpha, \sin \alpha)$, then ( )
A. $a^{2}+b^{2} \leqslant 1$
B. $a^{2}+b^{2} \geqslant 1$
C. $\frac{1}{a^{2}}+\frac{1}{b^{2}} \leqslant 1$
D. $\frac{1}{a^{2}}+\frac{1}{b^{2}} \geqslant 1$
According to the conditions and Cauchy's inequality, we have
$$1=\left(\frac{\cos \alpha}{a}+\frac{\sin \alpha}{b}\right)^{2} \leqslant\left(\cos ^{2} \alpha+\sin ^{2} \alpha\right) \cdot\left(\frac{1}{a^{2}}+\right.$$
$\left.\frac{1}{b^{2}}\right)$,
so $\frac{1}{a^{2}}+\frac{1}{b^{2}} \geqslant 1$. Therefore, the answer is D.