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  4. In a Δ A B C, Sigma(b+c) tan (A/2) tan ((B-C/2)) is equal to

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Q. In a $\Delta A B C, \Sigma(b+c) \tan \frac{A}{2} \tan \left(\frac{B-C}{2}\right)$ is equal to

EAMCETEAMCET 2005

Solution:

$\Sigma(b+c) \tan \frac{A}{2} \tan \left(\frac{B-C}{2}\right) $
$\therefore (b+c) \tan \frac{A}{2} \tan \left(\frac{B-C}{2}\right) $
$ =(b+c) \cdot \frac{(b-c)}{(b+c)} \cot \frac{A}{2} \tan \frac{A}{2} $
$=b-c $
$ \therefore \Sigma(b+c) \tan \frac{B-C}{2} \tan \frac{A}{2} $
$ = b-c+c-a+a-b=0$