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If $b+c=3 a$, then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :
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$\cot \frac{B}{2} \cdot \cot \frac{C}{2}$
$=\sqrt{\frac{s(s-b)}{(s-c)(s-a)}} \times \sqrt{\frac{s(s-c)}{(s-a)(s-b)}}$
$=\frac{s}{s-a}=\frac{2 a}{2 a-a}=2 \quad(\because b+c=3 a)$
$=\sqrt{\frac{s(s-b)}{(s-c)(s-a)}} \times \sqrt{\frac{s(s-c)}{(s-a)(s-b)}}$
$=\frac{s}{s-a}=\frac{2 a}{2 a-a}=2 \quad(\because b+c=3 a)$
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