A) \[\frac{a}{a+b+c}\]
B) \[\frac{c}{a+b+c}\]
C) \[\frac{2a}{a+b+c}\]
D) \[\frac{2b}{a+b+c}\]
E) \[\frac{2c}{a+b+c}\]
Correct Answer: E
Solution :
\[\frac{\cot \frac{A}{2}\cot \frac{B}{2}-1}{\cot \frac{A}{2}\cot \frac{B}{2}}=1-\tan \frac{A}{2}\tan \frac{B}{2}\] \[=1-\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\sqrt{\frac{(s-a)(s-c)}{s(s-b)}}\] \[=1-\frac{s-c}{s}=\frac{c}{s}\] \[=\frac{2c}{a+b+c}\]
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