Question

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

Answer 1

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