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

Solution:

Given: Equation is cos 2x - cos x = 0

We know that, cos 2x = cos²x - sin²x

⇒ cos²x - sin²x - cos x = 0

From trigonometric identity,

cos²x + sin²x = 1

⇒ sin²x = 1 - cos²x

Thus,

cos²x - (1 - cos²x) - cos x = 0

2cos²x - cos x - 1 = 0

This resembles a quadratic equation,

Let cos x = a

2a² - a -1 = 0

On solving, we get,

(2a + 1)(a - 1) = 0

a = 1 or -1/2

cosx = 1 implies x = 0

i.e cos 0 = 1 since the interval is [0, 2π]

cos x = -1/2 implies that x = 2π/3, 4π/3

⇒ cos 2π/3 = -1/2

⇒ cos 4π/3 = -1/2

Therefore, the solution of the equation is x = 2π/3, 4π/3, 0.

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

Summary:

All solutions to the equation cos 2x - cos x = 0 in the interval [0, 2π] is x = 2π/3, 4π/3, 0.