已知 $\frac{\cos \alpha}{1+\sin \alpha}=\sqrt{3}$, 则 $\frac{\cos \alpha}{\sin \alpha-1}$ 的值为 ( )
由题意得, $\sin ^{2} \alpha+\cos ^{2} \alpha=1$,那么 $\frac{\cos \alpha}{1+\sin \alpha}=\frac{1-\sin \alpha}{\cos \alpha}=\sqrt{3}, \frac{\cos \alpha}{\sin \alpha-1}=\frac{\sqrt{3}}{3}$, 故答案选 B.