question_answer

In an isosceles right angled triangle ABC, a  value of \[\tan \left( \frac{A}{2} \right)+\tan \left( \frac{B}{2} \right)+\tan \left( \frac{C}{2} \right)\] is

A)  \[\sqrt{2}-1\]

B)  \[2\sqrt{2}\]

C)  \[2\sqrt{2}-1\]

D)  \[2\sqrt{2}+1\]

Correct Answer: C

Solution :

Given, an isosceles right angled triangle ABC. Then,   \[\angle A=\angle C={{45}^{o}}\] and \[\angle B={{90}^{o}}\] Now, \[\tan \left( \frac{A}{2} \right)+\tan \left( \frac{B}{2} \right)+\tan \left( \frac{C}{2} \right)\] \[=\tan \left( \frac{{{45}^{o}}}{2} \right)+\tan \left( \frac{{{90}^{o}}}{2} \right)+\tan \left( \frac{{{45}^{o}}}{2} \right)\] \[=\sqrt{2}-1+1+\sqrt{2}-1\] \[\left[ \because \,\,\,\tan \left( {{22}^{o}}\frac{1}{2} \right)=\sqrt{2}-1 \right]\] \[=2\sqrt{2}-1\]