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Q)

If sides of $\bigtriangleup ABC \;(a, b, c)$ are in AP, $cot\;{\large\frac{c}{2}}$ equals

$(a)\;3\;tan\;\large\frac{B}{2}\qquad(b)\;3\;tan\;\large\frac{A}{2}\qquad(c)\;3\;cot\;\large\frac{B}{2}\qquad(d)\;3\;cot\;\large\frac{A}{2}$

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A)
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Answer : (b) $3\;tan\;\large\frac{A}{2}$
Explanation : a , b , c are in AP
$2b=a+c$
$a+b+c = 3b$
$3b=2s$
$cot\;\large\frac{c}{2}=\sqrt{\large\frac{s(s-c)}{(s-a)(s-b)}}$
$=\sqrt{\large\frac{2s(2s-2c)}{(2s-2a)(2s-2b)}}$
$=\sqrt{\large\frac{3b(2s-2c)}{(2s-2a)b}}$
$=3\;\sqrt{\large\frac{b(s-c)}{3b(s-a)}}$
$=3\;\sqrt{\large\frac{(3b-2b)(s-c)}{3b(s-a)}}$
$=3\;\sqrt{\large\frac{(2s-2b)(s-c)}{2s(s-a)}}$
$=3\;\sqrt{\large\frac{(s-b)(s-c)}{s(s-a)}}$
$=3\;tan\;\large\frac{A}{2}\;.$
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