Question

If $A+B+C=\pi$, prove that $\tan \left(A/2\right)\tan \left(B/2\right)+\tan \left(B/2\right)\tan \left(C/2\right)+\tan \left(C/2\right)\tan \left(A/2\right)=1$.

Answer 1

$\because A+B+C=\pi$,
$\therefore A/2+B/2+C/2=\pi /2$
$\therefore \tan (\frac{A}{2}+\frac{B}{2}+\frac{C}{2})=\tan \frac{\pi }{2}=\infty$.
or $\frac{S_1-S_3}{1-S_2}=\infty$
$\therefore 1-S_2=0$ or $S_2=1$
or $\tan \frac{A}{2}\tan \frac{B}{2}+\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}=1$.