$A + B + C = 2 s \to s i n S + s i n (S - A) + s i n (S - B) - s i n (S - C) =$

4 c o s \frac{A}{2} . c o s \frac{B}{2} . c o s \frac{C}{2}

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4 c o s \frac{A}{2} . c o s \frac{B}{2} . s i n \frac{C}{2}

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4 c o s \frac{A}{2} . s i n \frac{B}{2} . c o s \frac{C}{2}

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4 s i n \frac{A}{2} . s i n \frac{B}{2} . s i n \frac{C}{2}

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Solution

The correct option is A $4 c o s \frac{A}{2} . c o s \frac{B}{2} . c o s \frac{C}{2}$

$A + B + C = 2 S$
$sin (s) + sin (S - A) + sin (S - B) - sin (S + C)$
$sin (S) - sin (S - C) + sin (S - A) + sin (S - B)$
$2 sin (\frac{C}{2}) cos (2 \frac{- c}{2}) + 2 sin (2 S - (A + B)) cos (\frac{B A}{2})$

$Put 2 S = A + B + C$
$2 sin (\frac{C}{2}) cos (\frac{A + B}{2}) + 2 sin (\frac{C}{2}) cos (\frac{B - A}{2})$
$2 sin (\frac{C}{2}) [cos (\frac{A + B}{2}) + cos (\frac{B - A}{2})]$
$2 sin (\frac{C}{2}) 2 cos \frac{B}{2} cos A = 4 sin (\frac{C}{2}) cos (\frac{A}{2}) cos (\frac{B}{2})$

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