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: Half Angle Formula
$\cot \frac{B}{2} = \sqrt{\frac{s(s-b)}{(s-a)(s-c)}}$ and $\cot \frac{C}{2} = \sqrt{\frac{s(s-c)}{(s-a)(s-b)}}$.
Step 2: Product
$\cot \frac{B}{2} \cdot \cot \frac{C}{2} = \sqrt{\frac{s^2(s-b)(s-c)}{(s-a)^2(s-c)(s-b)}} = \frac{s}{s-a}$.
Step 3: Use Condition
$s = \frac{a+b+c}{2} = \frac{a+3a}{2} = 2a$.
Value $= \frac{2a}{2a-a} = \frac{2a}{a} = 2$.
Final Answer: (C)