A) \[3\tan \frac{A}{2}\]
B) \[3\tan \frac{B}{2}\]
C) \[3\cot \frac{A}{2}\]
D) \[2\cot \frac{B}{2}\]
Correct Answer: A
Solution :
Since, the sides of the \[\Delta \,\,ABC\] are in\[AP.\] \[\therefore \] \[2b=a+c\] \[\Rightarrow \] \[3b=a+b+c=2s\] Now, \[\cot \frac{C}{2}=\sqrt{\frac{s(s-c)}{(s-a)(2s-2b)}}\] \[=\sqrt{\frac{2s(s-c)}{(s-a)(2s-2b)}}\] \[=\sqrt{\frac{3b(s-c)}{(s-a)(3b-2b)}}\] \[=\sqrt{\frac{3(s-c)}{s-a}}\] \[=3\sqrt{\frac{(s-c)}{3(s-a)}}\] \[\Rightarrow \] \[\cot \frac{C}{2}=3\sqrt{\frac{b(s-c)}{3b(s-a)}}\] \[\cot \frac{C}{2}=3\sqrt{\frac{(2s-2b)(s-c)}{2s(s-a)}}\]\[\left[ \begin{matrix} \because \,\,3b=2s \\ \Rightarrow b+2b=2s \\ \Rightarrow b=2s-2b \\ \end{matrix} \right.\] \[\Rightarrow \] \[\cos \frac{C}{2}=3\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\] \[=3\tan \frac{A}{2}\]
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