Please verify this identity,2cos ^2x+2 root over 2sinx=3

Please verify this equation,4sin^2x-(2+2 root over 3)sinx-root over 3=0

It's very difficult trying to understand what you're asking - what does "root over 3" mean? I think you may mean this:

4 \sin^2x - (2-2 \sqrt 3)\sin x- \sqrt 3 = 0

but if so then the identity is incorrect.