RHS = $(\sin A + \sin B)(\sin A – \sin B)$
$ = 2\sin \frac{{A + B}}{2}\cos \frac{{A – B}}{2}.2\cos \frac{{A + B}}{2}\sin \frac{{A – B}}{2}$
$ = 2\sin \frac{{A + B}}{2}\cos \frac{{A + B}}{2}.2\sin \frac{{A – B}}{2}\cos \frac{{A – B}}{2}$
$ = \sin (A + B).\sin (A – B)$ = LHS