Question

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

cos(2x) - cos(x)= 0​

Answer 1

Explanation:

Answer 2

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