Question

Prove: $\sin A+\sin B+\sin C=4\cos \left(\frac{A}{2}\right)\cos \left(\frac{B}{2}\right)\cos \left(\frac{C}{2}\right)$.

Answer 1

L.H.S.=$\sin A+\sin B+\sin C=4\cos \left(\frac{A}{2}\right)\cos \left(\frac{B}{2}\right)\cos \left(\frac{C}{2}\right)$

$=2\sin \left(\frac{A}{2}\right)\cos \left(\frac{A}{2}\right)+2\sin \left\{\right(\frac{B+C}{2}\left)\right\}\cos \left\{\right(\frac{B-C}{2}\left)\right\}$

$=2\sin \left(\frac{A}{2}\right)\cos \left(\frac{A}{2}\right)+2\cos \left(\frac{A}{2}\right)\cos \left\{\right(\frac{B-C}{2}\left)\right\}$

$=2\cos \left(\frac{A}{2}\right)\left[\sin \right(\frac{A}{2})+\cos \{\left(\frac{B-C}{2}\right)\left\}\right]$

$=2\cos \left(\frac{A}{2}\right)\left[\cos \right\{\left(\frac{B+C}{2}\right\}+\cos \left\{\right(\frac{B-C}{2}\left)\right\}]$

$=2\cos \left(\frac{A}{2}\right)\left[2\cos \right(\frac{B}{2}\left)\cos \right(\frac{C}{2}\left)\right]$

$=4\cos \left(\frac{A}{2}\right)\cos \left(\frac{B}{2}\right)\cos \left(\frac{C}{2}\right)$.