Question

Доказать тождество
(sinA+sin3A+sin5A)/(cosA+cos3A+cos5A)=tg3A

Answer 1

$\displaystyle \frac{\sin \alpha +\sin3\alpha+\sin5\alpha}{\cos\alpha+\cos3\alpha+\cos5\alpha} = \frac{2\sin3\alpha\cos2\alpha+\sin3\alpha}{2\cos2\alpha\cos3\alpha+\cos3\alpha} =\\ \\ \\ = \frac{\sin3\alpha(2\cos2\alpha+1)}{\cos3\alpha(2\cos2\alpha+1)}= \frac{\sin3\alpha}{\cos3\alpha} =tg3\alpha$