Question

Solve: $2(\sin x-\cos 2x)-\sin 2x(1+2\sin x)+2\cos x=0$.

Answer 1

Given, $2(\sin x-\cos 2x)-\sin 2x(1+2\sin x)+2\cos x=0$

$2\{\sin x-(1-2{\sin }^{2}x\left)\right\}-\left\{2\sin x\cos x\right(1+2\sin x\left)\right\}+2\cos x=0$

............. $[\because \sin 2x=2\sin x\cos x]and(\because 1-cos2x=2si{n}^{2}x)$

$2\{\sin x-(1-2{\sin }^{2}x\left)\right\}+2\cos x\{1-\sin x(1+2\sin x\left)\right\}=0$

or $(2{\sin }^{2}x+\sin x-1)(1-\cos x)=0$

or $(\sin x+1)(2\sin x-1)(1-\cos x)=0$

$\therefore \sin x=-1,\cos x=1,\sin x=\frac{1}{2}$

$\therefore x=2n\pi +\frac{3\pi }{2},x=2n\pi ,n\pi +(-1{)}^n\frac{\pi }{6}$