若 $4 \sin \alpha-3 \cos \alpha=0$, 则 ${ }^{\frac{1}{\cos ^{2} \alpha+2 \sin 2 \alpha}}$ 的值为 ( )
$$
\frac{1}{\frac{1}{\cos ^{2} \alpha+2 \sin ^{2} \alpha}}=\frac{\sin ^{2} \alpha+\cos ^{2} \alpha}{\cos ^{2} \alpha+4 \sin \alpha \cos \alpha}=\frac{\left(\frac{3}{4} \cos \alpha\right)^{2}+\cos ^{2} \alpha}{\cos ^{2} \alpha+4\left(\frac{3}{4} \cos \alpha\right) \cos \alpha}=\frac{\frac{25}{16}}{4}=\frac{25}{64}
$$

故选 D