$\sin (A + B)\sin (A – B) = {\sin ^2}A – {\sin ^2}B \, Proof$

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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