Verify that each equation is an identity.

sin 4y= 4sinycosy - 8sin^3ycosy   

             factor out 4sinycosy  on the right

         =  [4sinycosy]  [ 1 - 2sin^2 y]

         =   [2sin2y ] [ cos2y ]

             Remember that sin2y  = 2 siny cosy........so,  2 sin2y cos2y =

            sin 4y

And the right side = the left side

I think you might mean this:

sin2x/2sinx = cos^2[x\2]-sin^2[x\2]   

                  [  remember,  cos^2x - sin^2x =  cos2x ..... so cos^2(x/2) - sin^2(x/2)  = cosx ]

2sinxcosx / 2sinx    =  cosx

cosx   = cosx