Question

Solve the following equation:
$(2\sin x-\cos x)(1+\cos x)={\sin }^{2}x.$

Answer 1

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

$2\sin x+2\sin x\cos x-\cos x-{\cos }^{2}x={\sin }^{2}x$

$2\sin x+2\sin x\cos x-\cos x-1=0$

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

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

Either, $(1+\cos x)=0\cos x=-1,x=(2n+1)\pi .$

Or, $2\sin x-1=0,\sin x=\frac{1}{2},x=n\pi +{(-1)}^n\frac{\pi }{6}$