sin(x+y) = sinx*cosy + cosx*siny

cos(x-y) = cosx*cosy + sinx*siny

sin(x+y)*cos(x-y) = sinxcosy*cosxcosy + sinxcosy*sinxsiny + cosxsiny*cosxcosy + cosxsiny*sinxsiny

= sinxcosx*cos^2y + sinycosy*sin^2x + sinycosy*cos^2x + sinxcosx*sin^2y

So

2sin(x+y)cos(x-y) = 2[sinxcosx(sin^2y + cos^2y) + sinycosy(sin^2x + sin^2y)]

= 2sinxcosx + 2sinycosy

= sin2x + sin2y