Question

Solve $\sin 6\theta =\sin 4\theta -\sin 2\theta$

Answer 1

Given, $\sin 6\theta =\sin 4\theta -\sin 2\theta \Rightarrow \sin 6\theta +\sin 2\theta =\sin 4\theta$
$\Rightarrow 2\sin \left(\frac{6\theta +2\theta }{2}\right)\cos \left(\frac{6\theta -2\theta }{2}\right)=\sin 4\theta$
$\Rightarrow 2\sin 4\theta \cos 2\theta -\sin 4\theta =0\Rightarrow \sin 4\theta \left(2\cos 2\theta -1\right)=0$
$\Rightarrow \sin 4\theta =0$ or $\cos 2\theta =\frac{1}{2}$
$\Rightarrow 4\theta =n\pi$ or $2\theta =2n\pm \frac{\pi }{3}$
$\Rightarrow \theta =\frac{n\pi }{4}$ or $\theta =n\pi \pm \frac{\pi }{6}$