Question

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

Answer 1

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,$\frac{\sin 3x}{\sin x}-\frac{1}{3}=2$
$\Rightarrow \frac{\sin 3x}{\sin x}=2+\frac{1}{3}=\frac{7}{3}$