Question

If $A+B+C=2S$, show that $\sin (S-A)+\sin (S-B)+\sin (S-C)-\sin S$=$4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$.

Answer 1

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