Triangles And Properties Of Triangle Question 37
Question: If $ A+B+C=\pi $ then $ \Sigma \tan \frac{A}{2}\tan \frac{B}{2}= $
Options:
A) 1
B) -1
C) 2
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] since $ A+B+C=\pi $
$ \therefore \frac{A}{2}+\frac{B}{2}+\frac{C}{2}=\frac{\pi }{2}\Rightarrow \frac{A}{2}+\frac{B}{2}=\frac{\pi }{2}-\frac{C}{2} $
$ \therefore \tan ( \frac{A}{2}+\frac{B}{2} )=\tan ( \frac{\pi }{2}-\frac{C}{2} )=\cot \frac{C}{2} $
$ \Rightarrow \frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{1-\tan \frac{A}{2}\tan \frac{B}{2}}=\frac{1}{\tan \frac{C}{2}} $
$ \Rightarrow \tan \frac{A}{2}\tan \frac{C}{2}+\tan \frac{B}{2}\tan \frac{C}{2}=1-\tan \frac{A}{2}\tan \frac{B}{2} $
$ \Rightarrow \tan \frac{A}{2}\tan \frac{B}{2}+\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}=1 $
BETA
