Question

Solve : $\sin 2x-\sin 4x+\sin 6x=0$.

Answer 1

$\Rightarrow (\sin 2x+\sin 6x)-\sin 4x=0$

$\Rightarrow 2\sin \left(\frac{2x+6x}{2}\right)\cos \left(\frac{2x-6x}{2}\right)-\sin 4x=0$

$\Rightarrow 2\sin 4x\cos 2x-\sin 4x=0$

$\Rightarrow \sin 4x(2\cos 2x-1)=0$

$\Rightarrow \sin 4x=0,2\cos 2x-1=0$

$\Rightarrow \sin 4x=0,\cos 2x=\frac{1}{2}$

$\therefore 4x=n\pi ,2x=2n\pi \pm \frac{\pi }{3}$

$\therefore x=\frac{n\pi }{4},n\pi \pm \frac{\pi }{6}$ where $n\in Z$