Question

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

Answer 1

Answer: 7

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