Question

which double angle or half angle identity would you use to verify that sin2x/2sin2x=cot x

Answer 1

erm, you wrote that out extremely poorly. I'm guessing you meant:
\[\frac{ \sin^2(x) }{ 2\sin(2x) } = \cot(x)\]
In that case, the only double angle in that formula is sin(2x)
sin(2x) = 0.5*(cosx)(sinx)