Find all solutions to the equation

Question

https://imgur.com/gallery/bpjrIVL

Answer 1

cos2x = cosx; 2cos²x - cosx - 1 = 0; (2cosx + 1)(cosx - 1) = 0; 2cosx + 1 = 0, cos x = - 1/2, x = 2π/3 or x = 4π/3.

cosx - 1 = 0 cosx = 1, x = 0.

So on interval [0, 2π) this equation has 3 solutions: 0, 2π/3 and 4π/3.

Answer 2

cos 2x = cos x

2 cos² - 1 = cos x

2 cos² - cos x - 1 = 0

(2 cos x + 1)(cos x - 1) = 0

2 cos x = -1

cos x = -1/2

cos x = 1