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$1 - 2\cos 2x - \sqrt 3 \sin x + \cos x = 0$$\Leftrightarrow 1 - 2(2{\cos ^2}x - 1) + \cos x - \sqrt 3 \sin x = 0$$\Leftrightarrow - 4{\cos ^2}x + \cos x + 3 - \sqrt 3 \sin x = 0$$\Leftrightarrow (1 - \cos x)(4\cos x + 3) - \sqrt 3 \sin x = 0$$\Leftrightarrow 2{\sin ^2}\frac{x}{2}(4\cos x + 3) - 2\sqrt 3 \sin \frac{x}{2}c{\rm{os}}\frac{x}{2} = 0$$\Leftrightarrow \left [ \begin{gathered} 2\sin \frac{x}{2} = 0 (1) \\ \sin \frac{x}{2}(4\cos x + 3) - \sqrt 3 c{\rm{os}}\frac{x}{2} = 0 (2) \end{gathered} \right.$$(1) \Leftrightarrow \frac{x}{2} = k\pi \Leftrightarrow x = k2\pi (k \in Z)$$(2) \Leftrightarrow 4\cos x\sin \frac{x}{2} + 3\sin \frac{x}{2} - \sqrt 3 c{\rm{os}}\frac{x}{2} = 0$ $\Leftrightarrow 2\sin \frac{{3x}}{2} - 2\sin \frac{x}{2} + 3\sin \frac{x}{2} - \sqrt 3 c{\rm{os}}\frac{x}{2} = 0$$\Leftrightarrow \sin \frac{{3x}}{2} = \frac{{\sqrt 3 }}{2}c{\rm{os}}\frac{x}{2} - \frac{1}{2}\sin \frac{x}{2}$$\Leftrightarrow \sin \frac{{3x}}{2} = \sin \left( {\frac{\pi }{3} - \frac{x}{2}} \right)$$\Leftrightarrow \left [ \begin{gathered} \frac{{3x}}{2} = \frac{\pi }{3} - \frac{x}{2} + k2\pi \\ \frac{{3x}}{2} = \frac{{2\pi }}{3} + \frac{x}{2} + k2\pi \end{gathered} \right.$$\Leftrightarrow \left [ \begin{gathered} x = \frac{\pi }{6} + k\pi \\ x = \frac{{2\pi }}{3} + k2\pi \end{gathered} \right.$ $(k \in Z)$

$1 - 2\cos 2x - \sqrt 3 \sin x + \cos x = 0$$\Leftrightarrow 1 - 2(2{\cos ^2}x - 1) + \cos x - \sqrt 3 \sin x = 0$$\Leftrightarrow - 4{\cos ^2}x + \cos x + 3 - \sqrt 3 \sin x = 0$$\Leftrightarrow (1 - \cos x)(4\cos x + 3) - \sqrt 3 \sin x = 0$$\Leftrightarrow 2{\sin ^2}\frac{x}{2}(4\cos x + 3) - 2\sqrt 3 \sin \frac{x}{2}c{\rm{os}}\frac{x}{2} = 0$$\Leftrightarrow \left [ \begin{gathered} 2\sin \frac{x}{2} = 0 (1) \\ \sin \frac{x}{2}(4\cos x + 3) - \sqrt 3 c{\rm{os}}\frac{x}{2} = 0 (2) \end{gathered} \right.$$(1) \Leftrightarrow \frac{x}{2} = k\pi \Leftrightarrow x = k2\pi (k \in Z)$$(2) \Leftrightarrow 4\cos x\sin \frac{x}{2} + 3\sin \frac{x}{2} - \sqrt 3 c{\rm{os}}\frac{x}{2} = 0$ $\Leftrightarrow 2\sin \frac{{3x}}{2} - 2\sin \frac{x}{2} + 3\sin \frac{x}{2} - \sqrt 3 c{\rm{os}}\frac{x}{2} = 0$$\Leftrightarrow \sin \frac{{3x}}{2} = \frac{{\sqrt 3 }}{2}c{\rm{os}}\frac{x}{2} - \frac{1}{2}\sin \frac{x}{2}$$\Leftrightarrow \sin \frac{{3x}}{2} = \sin \left( {\frac{\pi }{3} - \frac{x}{2}} \right)$$\Leftrightarrow \left [ \begin{gathered} \frac{{3x}}{2} = \frac{\pi }{3} - \frac{x}{2} + k2\pi \\ \frac{{3x}}{2} = \frac{{2\pi }}{3} + \frac{x}{2} + k2\pi \end{gathered} \right.$$\Leftrightarrow \left [ \begin{gathered} x = \frac{\pi }{6} + k\pi \\ x = \frac{{2\pi }}{3} + k2\pi \end{gathered} \right.$ $(k \in Z)$