A) \[1/3\]
B) \[2/3\]
C) \[2/9\]
D) \[7/9\]
Correct Answer: D
Solution :
Since, \[A,\,\,\,B\] and \[C\] are the angles of a triangle \[\therefore \] \[A+B+C=\pi \] \[\Rightarrow \] \[\frac{A}{2}+\frac{B}{2}=\frac{\pi }{2}-\frac{C}{2}\] \[\Rightarrow \] \[\tan \left( \frac{A}{2}+\frac{B}{2} \right)=\tan \left( \frac{\pi }{2}-\frac{C}{2} \right)\] \[\Rightarrow \] \[\cot \frac{C}{2}=\frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}\tan \frac{B}{2}}\] \[=\frac{\frac{1}{3}+\frac{2}{3}}{1-\frac{1}{3}\times \frac{2}{3}}=\frac{\frac{3}{3}}{\frac{7}{9}}=\frac{9}{7}\] \[\Rightarrow \] \[\tan \frac{C}{2}=\frac{7}{9}\]
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