What are the angles for 2cos2x=-1?

cos 2x = - 1/2
2x = 120⁰ , 240⁰ , 480⁰ . 600⁰
x = 60⁰ , 120⁰ , 240⁰ , 300⁰

2cos(2x) = -1
∴ cos(2x) = -1/2
given cos(θ) = x then θ = 2πn ± cos⁻¹(x), where n∈ℤ [note n∈ℤ means n is a member of the set of integers]
∴ 2x = 2πn ± cos⁻¹(-1/2), where n∈ℤ
∴ 2x = 2πn ± 2π/3, where n∈ℤ
∴ x = πn ± π/3, where n∈ℤ
over the interval of 0 ≤ x < 2π
∴ x = π/3, 2π/3, 4π/3 and 5π/3

2cos2x = -1
x = -1/(2cos2)
I suspect you really meant 2cos(2x) or 2cos^2x but which?