Question

Доказать тождество cos3a/ cosa - sin3a/sina = -2

Answer 1

$\frac{\cos3\alpha}{\cos\alpha}-\frac{\sin3\alpha}{\sin\alpha}=-2\\\sin3\alpha=3\sin\alpha-4\sin^3\alpha\quad\quad\cos3\alpha=4\cos^3\alpha-3\cos\alpha\\\frac{\cos3\alpha}{\cos\alpha}-\frac{\sin3\alpha}{\sin\alpha}=\frac{4\cos^3\alpha-3\cos\alpha}{\cos\alpha}-\frac{3\sin\alpha-4\sin^3\alpha}{\sin\alpha}=\\=\frac{\cos\alpha(4\cos^2\alpha-3)}{\cos\alpha}-\frac{\sin\alpha(3-4\sin^2\alpha)}{\sin\alpha}=4\cos^2\alpha-3-3+4\sin^2\alpha=\\=4(\sin^2\alpha+\cos^2\alpha)-6=4-6=-2$

ЧТД