prove sin a+sin b+sin c =? 4cos ((a/2))cos ((b/2))cos ((c/2))


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Question Number 81760 by jagoll last updated on 15/Feb/20

$p r o v e$ $\sin a + \sin b + \sin c = ?$ $4 \cos (\frac{a}{2}) \cos (\frac{b}{2}) \cos (\frac{c}{2})$
Commented by jagoll last updated on 15/Feb/20

$t h a n k y o u$
Commented by john santu last updated on 15/Feb/20
$⇒sin a+sin b+sin c = 2sin (((a+b)/2))cos (((a−b)/2))+2sin ((c/2))cos ((c/2)) = 2sin (((a+b)/2))cos (((a−b)/2))+2sin (90^o −(((a+b))/2))cos (90^o −(((a+b))/2)) = 2sin (((a+b)/2))cos (((a−b)/2))+2cos (((a+b)/2))sin (((a+b)/2)) = 2sin (((a+b)/2)) {cos (((a−b)/2))+cos (((a+b)/2))}= 2sin (((a+b)/2))×2cos ((a/2))cos (−(b/2)) = 2sin (90^o −((c/2)))×2cos ((a/2))cos ((b/2)) = 4 cos ((a/2))cos ((b/2))cos ((c/2))$
$\Rightarrow \sin a + \sin b + \sin c =$ $2 \sin (\frac{a + b}{2}) \cos (\frac{a - b}{2}) + 2 \sin (\frac{c}{2}) \cos (\frac{c}{2}) =$ $2 \sin (\frac{a + b}{2}) \cos (\frac{a - b}{2}) + 2 \sin (90^{o} - \frac{(a + b)}{2}) \cos (90^{o} - \frac{(a + b)}{2}) =$ $2 \sin (\frac{a + b}{2}) \cos (\frac{a - b}{2}) + 2 \cos (\frac{a + b}{2}) \sin (\frac{a + b}{2}) =$ $2 \sin (\frac{a + b}{2}) {\cos (\frac{a - b}{2}) + \cos (\frac{a + b}{2})} =$ $2 \sin (\frac{a + b}{2}) \times 2 \cos (\frac{a}{2}) \cos (- \frac{b}{2}) =$ $2 \sin (90^{o} - (\frac{c}{2})) \times 2 \cos (\frac{a}{2}) \cos (\frac{b}{2}) =$ $4 \cos (\frac{a}{2}) \cos (\frac{b}{2}) \cos (\frac{c}{2})$
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