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