use an identity to solve the following equation on the interval 0,2π).\ncos 2x = cos x\nselect the correct…

use an identity to solve the following equation on the interval 0,2π).\ncos 2x = cos x\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. x=\n(use integers or fractions for any numbers in the expression. type an exact answer, using π as needed. use a comma to separate answers as needed.)\nb. there is no solution.

use an identity to solve the following equation on the interval 0,2π).\ncos 2x = cos x\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. x=\n(use integers or fractions for any numbers in the expression. type an exact answer, using π as needed. use a comma to separate answers as needed.)\nb. there is no solution.

Answer

Explanation:

Step1: Apply double - angle identity

Use the double - angle identity (\cos2x = 2\cos^{2}x - 1). The equation (\cos2x=\cos x) becomes (2\cos^{2}x - 1=\cos x).

Step2: Rearrange the equation

Rearrange to get a quadratic equation: (2\cos^{2}x-\cos x - 1 = 0). Let (t = \cos x), then (2t^{2}-t - 1=0).

Step3: Factor the quadratic equation

Factor (2t^{2}-t - 1=(2t + 1)(t - 1)). So, ((2\cos x+1)(\cos x - 1)=0).

Step4: Solve for (\cos x)

Set each factor equal to zero:

  • If (2\cos x+1 = 0), then (\cos x=-\frac{1}{2}). On the interval ([0,2\pi)), (x=\frac{2\pi}{3},\frac{4\pi}{3}).
  • If (\cos x - 1=0), then (\cos x = 1). On the interval ([0,2\pi)), (x = 0).

Answer:

(x = 0,\frac{2\pi}{3},\frac{4\pi}{3})