[latex]cos^{2}x+cos^{2} \frac{ \pi }{6}=cos^{2}(2x)+sin^{2} \frac{\pi}{3} [/latex]

[latex]\cos^2x+\cos^2\pi/6 = \cos^2(2x)+\sin^2\pi/3\\ \cos^2x+3/4 = \cos^2(2x)+3/4\\ \cos^2x = \cos^2(2x) \\ 2\cos^2x = 2\cos^2(2x) \\ 1+\cos(2x) = 1 + \cos(4x)\\ \cos(2x) = \cos(4x) \\ 4x = 2x + 2\pi n, n\in \mathbb{Z}\\ 2x = 2\pi n, n\in \mathbb{Z}\\ x = \pi n, n\in \mathbb{Z}[/latex]