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Решите задачу:

$\displaystyle \cot(\alpha)-\tan(\alpha)=\frac{\cos(\alpha)}{\sin(\alpha)}-\frac{\sin(\alpha)}{\cos(\alpha)}=\frac{\cos(\alpha)}{\sin(\alpha)}\frac{\cos(\alpha)}{\cos(\alpha)}-\frac{\sin(\alpha)}{\cos(\alpha)}\frac{\sin(\alpha)}{\sin(\alpha)}$
$\displaystyle=\frac{\cos(\alpha)^2}{\sin(\alpha)\cos(\alpha)}-\frac{\sin(\alpha)^2}{\sin(\alpha)\cos(\alpha)}=\frac{\cos(\alpha)^2-\sin(\alpha)^2}{\frac122\sin(\alpha)\cos(\alpha)}=\frac{\cos(2\alpha)}{\frac12\sin(2\alpha)};$
$\displaystyle\therefore\cot(\alpha)-\tan(\alpha)=\boxed{2\cot(2\alpha)}\phantom{.}.$