$3 \cdot 15$ Simplify: $\frac{\csc \alpha}{\sqrt{1+\operatorname{ctg}^{2} \alpha}}-\frac{\cos \alpha}{\sqrt{1-\sin ^{2} \alpha}}$.
(Shanghai Mathematical Competition, China, 1962)
[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 $k=0, \pm 1, \pm 2, \cdots$)