To solve for x in the equation -2cosx+2cos2x=0, we can use the trigonometric identity cos2x = 2cos^2x - 1 to rewrite the equation as:
-2cosx + 4cos^2x - 2 = 0
Next, we can rearrange the terms and factor out a 2 to obtain:
2cos^2x - cosx - 1 = 0
This is now a quadratic equation in terms of cosx. We can solve for cosx using the quadratic formula:
cosx = [1 ± sqrt(1 - 4(2)(-1))] / (2(2))
cosx = [1 ± sqrt(9)] / 4
cosx = (1/2) or (-1/2)
Now, we need to find the values of x that correspond to these values of cosx. We can use inverse trigonometric functions to do this:
cosx = 1/2 => x = π/3 + 2πn or x = 5π/3 + 2πn, where n is an integer.
cosx = -1/2 => x = 2π/3 + 2πn or x = 4π/3 + 2πn, where n is an integer.
Therefore, the solutions for x are:
x = π/3 + 2πn, x = 2π/3 + 2πn, x = 4π/3 + 2πn, x = 5π/3 + 2πn, where n is an integer.