sin(x+y)=sin(x)cos(y)+cos(x)sin(y); cos(x-y)=cos(x)cos(y)+sin(x)sin(y).
sin(x+y)cos(x-y)=
[sin(x)cos(y)+cos(x)sin(y)][cos(x)cos(y)+sin(x)sin(y)]=
sin(x)cos(x)cos2(y)+sin2(x)cos(y)sin(y)+cos2(x)sin(y)cos(y)+cos(x)sin2(y)sin(x)=
[sin(x)cos(x)cos2(y)+cos(x)sin2(y)sin(x)]+[sin2(x)cos(y)sin(y)+cos2(x)sin(y)cos(y)]=
sin(x)cos(x)(sin2(y)+cos2(y))+sin(y)cos(y)(sin2(x)+cos2(x))=
sin(x)cos(x)+sin(y)cos(y) QED.