Question

In a triangle \(ABC\), \[ \tan \frac{A}{2}\tan \frac{B}{2} +\tan \frac{B}{2}\tan \frac{C}{2} +\tan \frac{C}{2}\tan \frac{A}{2} = \]

• A. \(0\)
• B. \(1\)
• C. \(\frac{1}{2}\)
• D. \(\pi\)

Hint:

For a triangle, \[ \frac{A}{2}+\frac{B}{2}+\frac{C}{2}=\frac{\pi}{2}. \] Use the identity \[ x+y+z=\frac{\pi}{2} \Rightarrow \tan x\tan y+\tan y\tan z+\tan z\tan x=1. \]

Answer 1

The Correct Option is B

Step 1: Use the angle sum property of triangle.
In triangle \(ABC\), \[ A+B+C=\pi \] Dividing by \(2\), \[ \frac{A}{2}+\frac{B}{2}+\frac{C}{2}=\frac{\pi}{2} \]

Step 2: Use tangent identity.
If \[ x+y+z=\frac{\pi}{2}, \] then \[ \tan x\tan y+\tan y\tan z+\tan z\tan x=1 \] Here, \[ x=\frac{A}{2},\quad y=\frac{B}{2},\quad z=\frac{C}{2} \] Therefore, \[ \tan \frac{A}{2}\tan \frac{B}{2} +\tan \frac{B}{2}\tan \frac{C}{2} +\tan \frac{C}{2}\tan \frac{A}{2} =1 \]

Step 3: Final conclusion.
Therefore, \[ \boxed{1} \]