$\left\{x\in R:\left|\cos 2x\right|+2{\cos }^{2}x=2\right\} =$

$$\begin{gathered} \left|\cos 2\mathrm{x}\right|+2{\cos }^2\mathrm{x}=2 \\ \left|\cos 2\mathrm{x}\right|+2{\cos }^2\mathrm{x}-1=1 \\ \left|\cos 2\mathrm{x}\right|+\cos 2\mathrm{x}=1 \\ 2\cos 2\mathrm{x}=1 \Rightarrow \cos 2\mathrm{x}=\frac{1}{2} \\ 2\mathrm{x}=2\mathrm{n\pi }\pm \frac{\mathrm{\pi }}{3} \\ \mathrm{x}=\mathrm{n\pi }\pm \frac{\mathrm{\pi }}{6} \end{gathered}$$