Question

In a triangle ABC with usual notations, if $3a = b + c$, then $\cot \frac{B}{2} \cdot \cot \frac{C}{2} =$

• A. 1
• B. $\sqrt{2}$
• C. 2
• D. 3

Hint:

$\cot \frac{B}{2} \cot \frac{C}{2} = \frac{s}{s-a}$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Use the half-angle cotangent formula in terms of semi-perimeter $s$.
Step 2: Key Formula or Approach:
$\cot \frac{B}{2} \cdot \cot \frac{C}{2} = \sqrt{\frac{s(s-b)}{(s-a)(s-c)}} \cdot \sqrt{\frac{s(s-c)}{(s-a)(s-b)}} = \frac{s}{s-a}$.
Step 3: Detailed Explanation:
$2s = a + b + c = a + 3a = 4a \implies s = 2a$.
Value = $\frac{2a}{2a - a} = \frac{2a}{a} = 2$.
Step 4: Final Answer:
The value is 2.