Solve: $\sin 3x = \sin x$
$\sin 3x = \sin x$
i.e. $\;$ $3 \sin x - 4 \sin^3 x = \sin x$
i.e. $\;$ $2 \sin x - 4 \sin^3 x = 0$
i.e. $\;$ $2 \sin x \left(1 - 2 \sin^2 x\right) = 0$
i.e. $\;$ $2 \sin x \times \cos 2 x = 0$
$\implies$ $\sin x = 0$ $\;$ or $\;$ $\cos 2x = 0$
Now, when $\sin x = 0$ $\implies$ $x = n \pi$, $\;\;\;$ $n \in Z$;
when $\cos 2x = 0$ $\implies$ $\cos 2x = \cos \left(\dfrac{\pi}{2}\right)$
$\implies$ $2x = 2 m \pi \pm \dfrac{\pi}{2}$ $\;$ i.e. $\;$ $x = m \pi \pm \dfrac{\pi}{4}, \;\;\; m \in Z$