A) \[\pi /6\]
B) 3
C) \[2\]
D) 1
E) \[-1\]
Correct Answer: D
Solution :
Given that, \[A+B+C=\pi \] \[\therefore \]\[\tan \frac{A}{2}\tan \frac{B}{2}+\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}\] \[\Rightarrow \]\[\tan \frac{B}{2}\left( \tan \frac{A}{2}+\tan \frac{C}{2} \right)+\tan \frac{C}{2}\tan \frac{A}{2}\] \[\Rightarrow \]\[\tan \frac{B}{2}\left( \frac{\sin \frac{A}{2}\cos \frac{C}{2}+\sin \frac{C}{2}\cos \frac{A}{2}}{\cos \frac{A}{2}\cos \frac{C}{2}} \right)\] \[+\frac{\sin \frac{C}{2}\sin \frac{A}{2}}{\cos \frac{C}{2}\cos \frac{A}{2}}\] \[\Rightarrow \]\[\frac{\tan \left( \frac{B}{2} \right)\left\{ \sin \left( \frac{A+C}{2} \right) \right\}+\sin \frac{C}{2}\sin \frac{A}{2}}{cos\frac{A}{2}\cos \frac{C}{2}}\] \[\Rightarrow \]\[\frac{\sin (B/2)+\sin (C/2)\sin (A/2)}{\cos (A/2)\cos (C/2)}\] \[\Rightarrow \]\[\frac{\cos (A+C)/2+\sin (C/2)\sin (A/2)}{\cos (A/2)\cos (C/2)}\] \[\Rightarrow \]\[\frac{\begin{align} & \cos (A+C)+\cos (C/2)-\sin (A/2) \\ & \sin (C/2)+\sin (C/2)\sin (A/2) \\ \end{align}}{\cos (A/2)\cos (C/2)}\] \[\Rightarrow \]\[\frac{\cos (A/2).\cos (C/2)}{\cos (A/2).\cos (C/2)}=1\]
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