If A + B + C = 180^o , then A 2 B 2 = - Vidyadip

Question

If $A + B + C = {180^o},$ then $\sum {\tan \frac{A}{2}\tan \frac{B}{2} = } $

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Answer

b
(b) We have $A + B + C = {180^o}$

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

$\therefore$ $\cot \frac{A}{2} = \tan \left( {\frac{B}{2} + \frac{C}{2}} \right)$

==> $\frac{1}{{\tan \frac{A}{2}}} = \frac{{\tan \frac{B}{2} + \tan \frac{C}{2}}}{{1 - \tan \frac{B}{2}\tan \frac{C}{2}}}$

==> $1 - \tan \frac{B}{2}\tan \frac{C}{2} $

$= \tan \frac{A}{2}.\tan \frac{B}{2} + \tan \frac{A}{2}.\tan \frac{C}{2}$

$\tan \frac{A}{2}.\tan \frac{B}{2} + \tan \frac{B}{2}\tan \frac{C}{2} + \tan \frac{A}{2}\tan \frac{C}{2} = 1$

$i.e.$, $\sum {\tan \frac{A}{2}\tan \frac{B}{2} = 1} $.

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