How does cos(x)+2cos^2 (x) -1=0 become cos(x)=1/2 or -1?

Answer 1

You are looking at a quadratic with cosx as the variable
we could do this:
let k = cosx, then your equation becomes
k + 2k^2 - 1 = 0
2k^2 + x - 1 = 0
(2k - 1)(k + 1) = 0
k = 1/2 or k = -1
then cosx = 1/2, cosx = -1

your next step probably would be to solve it for x

Answer 2

Thank man. I would never get it without your help.