Step-by-step explanation:
[tex] \cos(2x) = \cos(x) [/tex]
[tex] \cos {}^{2} (x) - \sin {}^{2} (x) = \cos(x) [/tex]
[tex] \cos {}^{2} (x) - (1 - \cos {}^{2} (x) ) = \cos(x) [/tex]
[tex]2 \cos {}^{2} (x) - 1 = \cos(x) [/tex]
[tex]2 \cos {}^{2} (x) - \cos(x) - 1 = 0[/tex]
[tex]2 \cos {}^{2} (x) - 2 \cos(x) + \cos(x) - 1 = 0[/tex]
[tex]2 \cos(x) ( \cos(x) - 1) + 1( \cos(x) + 1)[/tex]
[tex](2 \cos(x) + 1)( \cos(x) - 1) = 0[/tex]
[tex]2 \cos(x) + 1 = 0[/tex]
[tex]2 \cos(x) = - 1[/tex]
[tex] \cos(x) = - \frac{1}{2} [/tex]
[tex]x = \frac{2\pi}{3} ,x = \frac{4\pi}{3} [/tex]
[tex] \cos(x) - 1 = 0[/tex]
[tex] \cos(x) = 1[/tex]
[tex]x = 0[/tex]
So our answer are
[tex]0, \frac{2\pi}{3} , \frac{4\pi}{3} [/tex]
