Question

Show that $\sum a(\sin B-\sin C)=0$.

Answer 1

Solution:-

$\sum a(\sin B-\sin C)=0$

$L.H.S.-$

$\sum a(\sin B-\sin C)=a(\sin B-\sin C)+b(\sin C-\sin A)+c(\sin A-\sin B)$

From $\sin$ rule, i.e.,

$[\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=k]$

$\sum a(\sin B-\sin C)=a(bk-ck)+b(ck-ak)+c(ak-bk)$

$\Rightarrow \sum a(\sin B-\sin C)=abk-ack+bck-abk+cak-cbk$

$\Rightarrow \sum a(\sin B-\sin C)=0=R.H.S.$

Hence proved.