Question

How do you find the general solutions for `cos2x = 4cosx - 3`?

Answer 1

Express `cos 2x` in terms of `cos^2 x`, then solve the resulting quadratic to find `cos x = 1`, hence `x = 2n pi` for any integer `n`.

Explanation:

`cos 2x = cos^2 x - sin^2 x = cos^2 x - (1 - cos^2 x)`

`= 2 cos^2 x - 1`

So our equation becomes:

`2 cos^2 x - 1 = 4 cos x - 3`

Subtract the right hand side from the left to get:

`0 = 2 cos^2 x - 4 cos x + 2`

`=2(cos^2 x - 2 cos x + 1)`

`=2(cos x - 1)(cos x - 1)`

So `cos x = 1`

So `x = 2n pi` where `n` is any integer.