Find all exact solutions on the interval [0, 2pi].

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

Using the double angle identity, rewrite the equation as

cos(2x) - cos(x) = (2 cos²(x) - 1) - cos(x)

==>   2 cos²(x) - cos(x) - 1 = 0

Factorize the left side:

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

Then

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

cos(x) = -1/2   or   cos(x) = -1

In the interval [0, 2π], we get

x = 2π/3   or   x = 4π/3   or   x = π

step-by-step explanation: