Решите уравнение: \(2\cdot \frac{\sin 3x - \sin x}{\cos 3x +\cos x}=\sqrt{3}\cdot(1-\tan^{2} x)\)

Решение

\(2\cdot \frac{\sin 3x - \sin x}{\cos 3x +\cos x}=\sqrt{3}\cdot(1-\tan^{2} x) \iff 2\cdot \frac{\cos 2x \sin x}{\cos 2x \cos x} = \sqrt{3}\cdot\frac{\cos 2x}{\cos^{2} x}\iff \tan 2x \cdot \frac{\cos x}{\cos x}=\sqrt{3}\iff \\ \iff 2x=\frac{\pi}{3}+\pi k, k\in Z \iff x=\frac{\pi}{6}+\frac{\pi k}{2}, k\in Z\)

Ответ: \(\frac{\pi}{6}+\frac{\pi k}{2}, k\in Z\).