$\cos 3x+\cos x=2\cos x(\sin 2x+1)$$\Leftrightarrow 2\cos 2x\cos x=2\cos x(\sin 2x+1)$
$\Leftrightarrow 2\cos x(\cos 2x-\sin 2x-1)=0$
$\Leftrightarrow \cos x=0$ hoặc $\cos 2x-\sin 2x-1=0$(1)
Từ (1) $\Rightarrow \cos x^{2}-\sin x^{2}=2\sin x\cos x+1$
$\Leftrightarrow -2\sin x(\sin x-\cos x)=0 $ $\Rightarrow \sin x=0 $ hoặc $ \sin x-\cos x=0$
$\Rightarrow 1-2\sin x\cos x=0$ $\Leftrightarrow \sin 2x=1$