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  4. If A, B, C, are angles of triangle A B C and tan (A/2), tan (B/2), tan (C/2) are in H.P., then the minimum value of cot (B/2) is

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Q. If $A, B, C$, are angles of $\triangle A B C$ and $\tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2}$ are in H.P., then the minimum value of $\cot \frac{B}{2}$ is

Sequences and Series

Solution:

$ \cot \frac{ A }{2} \cdot \cot \frac{ B }{2} \cdot \cot \frac{ C }{2}=\cot \frac{ A }{2}+\cot \frac{ B }{2}+\cot \frac{ C }{2}$
Since, $\cot \frac{A}{2}, \cot \frac{B}{2}, \cot \frac{C}{2}$ are in A.P. $\Rightarrow \cot \frac{A}{2} \cdot \cot \frac{B}{2} \cdot \cot \frac{C}{2}=3 \cot \frac{B}{2}$ $\cot \frac{ A }{2} \cdot \cot \frac{ C }{2}=3$
ApplyA.M. $\geq$ G.M.
$\frac{\cot \frac{ A }{2}+\cot \frac{ C }{2}}{2} \geq \sqrt{\cot \frac{ A }{2} \cot \frac{ C }{2}}$
Hence, $\cot \frac{B}{2} \geq \sqrt{3}$.