Question

Prove that :
$(b+c-a)\{\cot \left(B/2\right)+\cot \left(C/2\right)\}=2a\cot \left(A/2\right).$

Answer 1

$L.H.S=2(s-a)\frac{\sin \frac{B+C}{2}}{\sin \frac{B}{2}\sin \frac{C}{2}}$
$=\frac{2(s-a).\cos \left(A/2\right)}{{[\frac{(s-c)(s-a)}{ca}.\frac{(s-a)(s-b)}{ab}]}^{1/2}}$
$=\frac{2\cos \left(A/2\right)}{{\left[\frac{(s-a)(s-c)}{bc}\right]}^{1/2}}$
$=2a\frac{\cos \left(A/2\right)}{\sin \left(A/2\right)}=a\cot \frac{A}{2}$

LHS=RHS

Hence proved.