Доказать тождество: sinx+siny+sin(x-y)=4sin(x/2)cos(x/2)cos((x-y)/2)

Sinx+siny+sin(x-y)=4sin(x/2)cos(x/2)cos((x-y)/2);
sinx+siny+sin(x-y)=2sin((x+y)/2)·cos((x-y)/2)+2sin(x-y)/2·cos(x-y)/2=
=2cos((x-y)/2)·[sin(x+y)/2+sin(x-y)/2)]=
=2cos((x-y)/2)·2sin((x+y+x-y)/4)·cos((x+y-x+y)/4)=
=4cos((x-y)/2)·sin(x/2)·cos(y/2)