Sinx+siny+sin (x+y) = 4sinx/2cosx/2cosx-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)