Question

How do you solve `cos2x= cosx` from 0 to 2pi?

Answer 1

`0, (2pi)/3, (4pi)/3, 2pi`

Explanation:

Use the identity: `cos 2x = 2cos^2 x - 1`. The given equation
transforms to:
`2cos^2 x - cos x - 1 = 0`.
Solve this quadratic equation for cos x.
Since a + b + c = 0, use shortcut. There are 2real roots:
cos x = 1 and `cos x = c/a = - 1/2`.
a. cos x = 1 --> x = 0 or `x = 2pi`
b. `cos x = - 1/2` ---> `x = +- (2pi)/3`
The co-terminal to arc `- (2pi)/3` --> arc `(4pi)/3`

Answers for `(0, 2pi)`:
`0, (2pi)/3, (4pi)/3, 2pi`