Using the double angle identity, rewrite the equation as
cos(2x) - cos(x) = (2 cos²(x) - 1) - cos(x)
==> 2 cos²(x) - cos(x) - 1 = 0
Factorize the left side:
(2 cos(x) + 1) (cos(x) - 1) = 0
Then
2 cos(x) + 1 = 0 or cos(x) - 1 = 0
cos(x) = -1/2 or cos(x) = -1
In the interval [0, 2π], we get
x = 2π/3 or x = 4π/3 or x = π
