If-in-a-ABC-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-

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} \dots \dots (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} \dots \dots (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 \dots \dots (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!