|(cot^2)2x + 8 sqrt(-cot2x) - 3|=|(cot^2)2x - 8 sqrt(-cot2x) - 3|

I think this equation leads to either:

(a) the midddle term is 0, so that -cot(2x) =0, or
(b) the sum of the other two terms is 0, so that cot^2(2x)-3 = 0. This leads to two solutions, but since -cot(2x) must be positive (so that the middle term with the square root is valid) only one of those solutions is possible.

From this you get a series of values for x that satisfy either (a) or (b). Can you go from there?