Question

If A, B, C, are angles of $\Delta ABC$ 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

• A. $-\sqrt{3}$
• B. $\sqrt{3}$
• C. $-2\sqrt{3}$
• D. $2\sqrt{3}$

Answer 1

The correct option is B $\sqrt{3}$

$\tan \frac{A}{2}=a\tan \frac{B}{2}=b\tan \frac{C}{2}$

$\Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c}$...(1)

$A+B+C=\pi \Rightarrow \frac{A}{2}+\frac{B}{2}+\frac{C}{2}=\frac{\pi }{2}$

$\tan (\frac{A}{2}+\frac{C}{2})=\cot \frac{B}{2}=\frac{1}{b}$...(2)

From (1) and (2)

$\tan \frac{A}{2}\times \tan \frac{C}{2}=\frac{1}{3}$

$a+c=\frac{2}{3b}$

$\cot \frac{B}{2}\ge \sqrt{3}$

$\therefore \cot \frac{B}{2}=\sqrt{3}$