$\cos 2x+\sqrt{3}\cos x=2\Rightarrow 2{{\cos }^{2}}x-1+\sqrt{3}\cos x-2=0$  $2{{\cos }^{2}}x+\sqrt{3}\cos x-3=0\Rightarrow \cos x=\frac{-\sqrt{3}\pm \sqrt{3+24}}{2\times 2}=\frac{-\sqrt{3}\pm 3\sqrt{3}}{4}\Rightarrow \left\{ \begin{matrix}    \cos x=\frac{\sqrt{3}}{2}\Rightarrow x=2k\pi \pm \frac{\pi }{6}  \\    \cos x=-\sqrt{3}  \\ \end{matrix} \right.$