If in a ΔABC, tan(A/2), tan(B/2) and tan(C/2) are in HP, then the minimum value of cot(B/2) is greater than (1) 2(√3) (2) ((√3)/2) (3) (√3) (4) (1/(√3))


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Question Number 20785 by Tinkutara last updated on 02/Sep/17

$If in a Δ A B C, \tan \frac{A}{2}, \tan \frac{B}{2} and \tan \frac{C}{2}$ $are in HP, then the minimum value of$ $\cot \frac{B}{2} is greater than$ $(1) 2 \sqrt{3}$ $(2) \frac{\sqrt{3}}{2}$ $(3) \sqrt{3}$ $(4) \frac{1}{\sqrt{3}}$
	Answered by ajfour last updated on 03/Sep/17

	$l e t u s c a l l \tan \frac{A}{2} = x, \tan \frac{B}{2} = y,$ $\tan \frac{C}{2} = z .$ $g i v e n x, y, z a r e i n H . P .$ $\Rightarrow y = \frac{2 x z}{x + z} . . . . . . (i)$ $\tan (\frac{A}{2} + \frac{B}{2} + \frac{C}{2}) = \frac{Σ \tan \frac{A}{2} - Π \tan \frac{A}{2}}{1 - Σ \tan \frac{A}{2} \tan \frac{B}{2}}$ $\Rightarrow 1 - Σ x y = 0$ $o r y (x + z) + x z = 1$ $y = \frac{1 - x z}{x + z} . . . . . . (i i)$ $f r o m (i) a n d (i i) i t f o l l o w s t h a t$ $2 x z = 1 - x z \Rightarrow x z = \frac{1}{3}$ $w e k n o w t h a t A . M . ⩾ G . M .$ ${(\frac{x + z}{2})}^{2} ⩾ x z . . . . . . (i i i)$ $y = \frac{2 x z}{x + z} [e q . (i)] a n d x z = 1 / 3$ $\Rightarrow x + z = \frac{2}{3 y}; u s i n g t h i s i n (i i i)$ ${(\frac{1}{3 y})}^{2} ⩾ \frac{1}{3} \Rightarrow \frac{1}{y} ⩾ \sqrt{3}$ $\Rightarrow \cot \frac{B}{2} ⩾ \sqrt{3} .$
	Commented by Tinkutara last updated on 03/Sep/17

	$Thank you very much Sir!$
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