2 cos2(𝜃) = 2 − cos(2𝜃)

Question

Use a Double- or Half-Angle Formula to solve the equation in the interval [0, 2𝜋). (Enter your answers as a comma-separated list.).

Answer 1

I'm assuming the left side is 2cos²(θ)

2cos²(θ) = 2 − cos(2θ)

Using cos(2θ) = 2cos²(x) – 1:

2cos²(θ) = 2 − (2cos²(x) – 1)

2cos²(θ) = 2 − 2cos²(x) + 1

2cos²(θ) + 2cos²(x) = 2 + 1

4cos²(x) = 3

cos²(x) = 3/4

cos(x) = ±√(3/4)

cos(x) = ±√3/2

x = π/6, 5π/6, 7π/6, 11π/6 (using the unit circle)