a,
$2\sin x\cos x-\sqrt3 \cos x=0$
$\to \cos x(2\sin x-\sqrt3)=0$
b,
$\sin2x+\sqrt3=2\cos x+\sqrt3\sin x$
$\to 2\sin x\cos x-2\cos x=\sqrt3\sin x-\sqrt3$
$\to 2\cos x(\sin x-1)=\sqrt3(\sin x-1)$
$\to (\sin x-1)(2\cos x-\sqrt3)=0$
c,
$\cos2x+(1+2\cos x)(\sin x-\cos x)=0$
$\to (\cos x-\sin x)(\cos x+\sin x)-(1+2\cos x)(\cos x-\sin x)=0$
$\to (\cos x-\sin x)(\cos x+\sin x-1-2\cos x)=0$
$\to (\sin x-\cos x)(\sin x-\cos x-1)=0$
d,
$\sin x(1+\cos2x)+\sin2x=1+\cos x$
$\to \sin x.2\cos^2x+2\sin x\cos x=\cos x+1$
$\to 2\sin x\cos x(\cos x+1)-(\cos x+1)=0$
$\to (\cos x+1)(2\sin x-1)=0$
