$3 \cdot 15$ Simplify: $\frac{\csc \alpha}{\sqrt{1+\operatorname{ctg}^{2} \alpha}}-\frac{\cos \alpha}{\sqrt{1-\sin ^{2} \alpha}}$.
[Solution] Original expression $=\frac{\csc \alpha}{|\csc \alpha|}-\frac{\cos \alpha}{|\cos \alpha|}$ $=\left\{\begin{array}{l}0, \text { when } 2 k \pi<\alpha<\left(2 k+\frac{1}{2}\right) \pi \text {; } \\ 2, \text { when }\left(2 k+\frac{1}{2}\right) \pi<\alpha<(2 k+1) \pi \text {; } \\ 0, \text { when }(2 k+1) \pi<\alpha<\left(2 k+\frac{3}{2}\right) \pi \text {; } \\ -2, \text { when }\left(2 k+\frac{3}{2}\right) \pi<\alpha<(2 k+2) \pi \text {; }\end{array}\right.$
(Where $k=0, \pm 1, \pm 2, \cdots$ )