می‌دانیم $\cos 2x=2{{\cos }^{2}}x-1$ پس $2{{\cos }^{2}}x=1+\cos 2x$.  $\begin{align}  & \cos 2x+2{{\cos }^{2}}x=0\Rightarrow \cos 2x+(1+\cos 2x)=0 \\  & \Rightarrow 2\cos 2x+1=0\Rightarrow \cos 2x=\frac{-1}{2} \\  & \Rightarrow \cos 2x=\cos \frac{2\pi }{3} \\  & \Rightarrow 2x=2k\pi \pm \frac{2\pi }{3}\Rightarrow x=k\pi \pm \frac{\pi }{3} \\ \end{align}$