If $A + B + C = π$ , prove that $(sin A + sin B + sin C) (- sin A + sin B + sin C) (sin A - sin B + sin C) (sin A + sin B - sin C) = 4 {sin}^{2} A {sin}^{2} B {sin}^{2} C$ .

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Solution

Then L.H.S. $= [4 cos \frac{A}{2} cos \frac{B}{2} cos \frac{C}{2}] [4 sin \frac{B}{2} sin \frac{C}{2} cos \frac{A}{2}]$ $\cdot [4 sin \frac{C}{2} sin \frac{A}{2} cos \frac{B}{2}] [4 sin \frac{A}{2} sin \frac{B}{2} cos \frac{C}{2}]$
$= 4 \times 64 ({sin}^{2} \frac{A}{2} {cos}^{2} \frac{A}{2})$ $() ()$
$= 4 {(2 sin \frac{A}{2} cos \frac{A}{2})}^{2}$ $() ()$
$= 4 {sin}^{2} A {sin}^{2} B {sin}^{2} C$ .

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