I also don't quite understand your question, but if you are solving for x. Firstly consider α = 2x, then times the domain by 2, i.e. [0, 4π] or 0 ≤ α ≤ 4π, then solve for cos α = 1 in this domain, the answer is: α = 0, 2π, 4π. i.e. 2x = 0, 2π, 4π ⇒ x = 0, π, 2π.
You can also solve this graphically, you first draw the graph of cos 2x, which is cos x that dilate horizontally by a factor of 1/2 in the domain of [0, 2π] (the graph contract), then you draw the line y = 1, and find the solutions for x.