solving trigo equation

cos x = cos 2x
cos x = 2 (cos x)^2 -1
2(cos x)^2 - cos x - 1=0
(2cos x +1)(cos x - 1)=0
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The skill is to make similar trigo terms, e.g. with cos x and cos 2x, you should convert cos 2x in terms of cos x.