It looks like the equation is
cos(2x) + cos(x) = 0
Use the double angle identity for cosine to write this as
(2 cos²(x) - 1) + cos(x) = 0
2 cos²(x) + cos(x) - 1 = 0
Factorize the left side:
(2 cos(x) - 1) (cos(x) + 1) = 0
Then either
2 cos(x) - 1 = 0 or cos(x) + 1 = 0
2 cos(x) = 1 or cos(x) = -1
cos(x) = 1/2 or cos(x) = -1
[x = cos⁻¹(1/2) + 2nπ or x = -cos⁻¹(1/2) + 2nπ]
… or [x = cos⁻¹(-1) + 2nπ]
(where n is any integer)
x = π/3 + 2nπ or x = -π/3 + 2nπ or x = π + 2nπ
We get the following solutions in the interval [0, 2π):
x = π/3 and x = π and x = 5π/3
