Question

If \( A + B + C = 180^\circ \), then the value of \( \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right) + \tan \left( \frac{B}{2} \right) \tan \left( \frac{C}{2} \right) + \tan \left( \frac{C}{2} \right) \tan \left( \frac{A}{2} \right) \) is

• A. 1
• B. -1
• C. -2
• D. 2

Hint:

For angles in a triangle or for angles summing to \( 180^\circ \), use trigonometric identities to simplify expressions and obtain the result.

Answer 1

The Correct Option is A

Step 1: Use the trigonometric identity.
We are given that \( A + B + C = 180^\circ \). Using trigonometric identities, we know that: \[ \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right) + \tan \left( \frac{B}{2} \right) \tan \left( \frac{C}{2} \right) + \tan \left( \frac{C}{2} \right) \tan \left( \frac{A}{2} \right) = 1 \]
Step 2: Apply the result.
Since \( A + B + C = 180^\circ \), we directly apply the known identity to conclude that the value of the expression is 1.
Step 3: Conclusion.
Thus, the value of the expression is 1.