Question

Упростить два выражения

Answer 1

$a) \frac{1}{tg\alpha } + \frac{\sin \alpha}{1 + \cos \alpha}= \frac{1}{\frac{\sin \alpha }{\cos \alpha } } + \frac{\sin \alpha}{1 + \cos \alpha}= \frac{\cos \alpha}{\sin \alpha } + \frac{\sin \alpha}{1 + \cos \alpha}=\frac{\cos \alpha(1+\cos \alpha)+\sin^2\alpha }{\sin \alpha (1+\cos \alpha )} =\\=\frac{\cos \alpha+\cos^2 \alpha+\sin^2\alpha }{\sin \alpha (1+\cos \alpha )} =\frac{\cos \alpha+1 }{\sin \alpha (1+\cos \alpha )} =\frac{1}{\sin \alpha }$

$b) \frac{\sin \alpha -\sin 3\alpha}{\cos 3\alpha + \cos \alpha }=\frac{2 \cos \frac{\alpha +3\alpha }{2} \sin\frac{\alpha -3\alpha }{2} }{2 \cos \frac{\alpha +3\alpha }{2}\cos\frac{\alpha -3\alpha }{2}} =\frac{\sin (-\alpha )}{\cos (-\alpha )} =tg(-\alpha )=-tg(\alpha)$