$sin A + sin B - sin C = 4 sin (A / 2) sin (B / 2) cos (C / 2)$ .

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Solution

$A + B + C = 180$
$sin A + sin B - sin C$
$= 2 sin (\frac{A + B}{2}) cos (\frac{A - B}{2}) - 2 sin (\frac{A + B}{2}) cos (\frac{A + B}{2})$
$= 2 sin (\frac{A + B}{2}) (cos (\frac{A - B}{2}) - cos (\frac{A + B}{2}))$
$= 2 cos \frac{C}{2} (2 sin (\frac{A - B - A - B}{2 \times 2}) sin (\frac{A + B + A - B}{2 \times 2}))$
$= 2 cos \frac{C}{2} \cdot 2 sin \frac{A}{2} sin \frac{B}{2}$
$= 4 sin \frac{A}{2} sin \frac{B}{2} cos \frac{C}{2}$

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