Help with solving trig question. Cos(Theta/2) - cos(theta) = 1

I'm having a hard time figuring out this trig question.
Solve for theta 0 Cos(Theta/2) - cos(theta) = 1

Use substitution for cosΘ based on fact that Θ = 2*Θ/2

cos(Θ/2) - (2cos²(Θ/2) - 1) = 1

cos(Θ/2) - 2cos²(Θ/2) + 1 = 1

2cos²(Θ/2) - cos(Θ/2) = 0

cos(Θ/2)[2cos(Θ/2) - 1] = 0

cos(Θ/2) = 0 or 2cos(Θ/2) - 1 = 0

Θ/2 = π/2, 3π/2 or cos(Θ/2) = 1/2 --> Θ/2 = π/3, 5π/3

Θ = π or 2π/3