Question

If $\frac{\cos 3 x}{\cos x}=\frac{1}{3}$ for some angle $x, 0 \leq x \leq \frac{\pi}{2}$, then the value of $3\left(\frac{\sin 3 x}{\sin x}\right)$ for same $x$, is

Answer 1

Consider, $\frac{\sin 3 x}{\sin x}-\frac{\cos 3 x}{\cos x}=\frac{\sin 3 x \cos x-\cos 3 x \sin x}{\sin x \cos x}$
$=\frac{\sin 2 x}{\sin x \cos x}=2 \cdot \frac{\sin 2 x}{\sin 2 x}=2$
so $\frac{\sin 3 x}{\sin x}-\frac{1}{3}=2$
or $ \frac{\sin 3 x}{\sin x}=2+\frac{1}{3}=\frac{7}{3}$