verifying a solution.
4cos^2 2x-2=0
4cos^2 2x-2=0
cos^2 (2x) = 1/2
cos (2x) = ± 1/√2
2x = 45° or 2x = 135° or 2x = 225° or 2x = 315°
x = 22.5°, 67.5°, 112.5°, 157.5, ...
the period of cos 2x is 180°, so adding 180° to any of the above answers will produce a new answer.
So there is an infinite number of solutions.