Question

Find all solutions to the equation in the interval [0, 2π). cos 4x - cos 2x = 0

Answer 1

See below in bold.

Explanation:

cos 4x = 2cos^2 2x - 1 = 2 (2 cos^2 x - 1)^2 - 1

and cos 2x = 2 cos^2 x - 1 so we have:

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

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

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

Let c = cos^2 x, then:

(2c - 1)^2 - c = 0

4c^2 - 4c + 1 - c = 0

4c^2 - 5c + 1 = 0

c = 0.25, 1

cos^2 x = 0.25 gives cos x = +/- 0.5

and cos^2 x = 1 gives cos x = +/- 1.

So for x = +/- 1 , x = 0, π.

For cos x = +/- 0.5, x = π/3, 2π/3, 4π3,5π/3.