Question

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

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

Hint:

This identity is specific to triangle angle half tangents. Memorize this for quick substitution in triangle trigonometry problems.

Answer 1

The Correct Option is B

In any triangle, the identity involving tangents of half angles is: \[ \tan\frac{A}{2} \tan\frac{B}{2} + \tan\frac{B}{2} \tan\frac{C}{2} + \tan\frac{C}{2} \tan\frac{A}{2} = 1 \] This is a standard trigonometric identity derived from: \[ \begin{align} \tan\frac{A}{2} \tan\frac{B}{2} + \tan\frac{B}{2} \tan\frac{C}{2} + \tan\frac{C}{2} \tan\frac{A}{2} = \frac{(s - b)(s - c)}{s(s - a)} + \cdots \Rightarrow \text{Sum } = 1 \]