3 csc^2 (2x) + 2 cosec 2x = 1-cot^2 (2x)

I am assuming you want to solve for x:-

Now cosec^2 2x = 1 - cot^2 2x so the equation becomes

3 cosec^2 (2x) + 2 cosec 2x = cosec^2 2x

2 cosec^2 2x + 2 cosec 2x = 0

2 cosec 2x (cosec 2x + 1) = 0

so 2 cosec 2x = 0 or cosec 2x = - 1

cosec 2x = 0 or cosec 2x = - 1

cosec 2x = 0 is indeterminate so we ignore the first one

cosec 2x = - 1 can be written as 1 / sin 2x = - 1

so sin 2x = - 1

giving 2x = 270 degrees

Therefore x = 135 degrees answer