$\sqrt 3 \sin x + 2\cos x - \cos 2x - 1 = 0$$ \Leftrightarrow \sqrt 3 \sin x + 2\cos x - {\cos ^2}x + {\sin ^2}x - 1 = 0$
$ \Leftrightarrow \sqrt 3 \sin x + {\sin ^2}x - ({\cos ^2}x - 2\cos x + 1) = 0$
$\Leftrightarrow \sin x(\sin x + \sqrt 3 ) - {(1 - \cos x)^2} = 0$
$ \Leftrightarrow 2\sin \frac{x}{2}\cos \frac{x}{2}(\sin x + \sqrt 3 ) - 2{\sin ^2}\frac{x}{2}(1 - \cos x) = 0$
$ \Leftrightarrow \sin \frac{x}{2} = 0 \vee \cos \frac{x}{2}(\sin x + \sqrt 3 ) - \sin \frac{x}{2}(1 - \cos x) = 0$
$*\sin \frac{x}{2} = 0 \Leftrightarrow x = k2\pi (k \in Z)$
$*\cos \frac{x}{2}(\sin x + \sqrt 3 ) - \sin \frac{x}{2}(1 - \cos x) = 0$
$ \Leftrightarrow \sin x\cos \frac{x}{2} + \cos x\sin \frac{x}{2} - \sin \frac{x}{2} + \sqrt 3 \cos \frac{x}{2} = 0$
$\Leftrightarrow \sin \frac{{3x}}{2} - 2\sin (\frac{x}{2} - \frac{\pi }{3}) = 0 (2)$
Đặt $t = \frac{x}{2} - \frac{\pi }{3} \Rightarrow \frac{{3x}}{2} = 3t + \pi \Rightarrow \sin \frac{{3x}}{2} = \sin (3t + \pi ) = - \sin 3t$
$(2) \Leftrightarrow - \sin 3t - 2\sin t = 0 \Leftrightarrow 4{\sin ^3}t - 5\sin t = 0$
$ \Leftrightarrow {\sin ^2}t = \frac{5}{4}(VL) \vee \sin t = 0 \Leftrightarrow t = \frac{x}{2} - \frac{\pi }{3} = h\pi \Leftrightarrow x = \frac{{2\pi }}{3} + h2\pi (h \in Z)$