show that If a^2 ,b^2 ,c^(2 ) are in A.P the cotA,cotB,cotC are also in A.P


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Question Number 46569 by scientist last updated on 28/Oct/18

$s h o w t h a t I f a^{2}, b^{2}, c^{2} a r e i n A . P t h e c o t A, c o t B, c o t C a r e$ $a l s o i n A . P$
	Answered by tanmay.chaudhury50@gmail.com last updated on 28/Oct/18

	$\frac{s i n A}{a} = \frac{s i n B}{b} = \frac{s i n c}{c} = k (s a y)$ $c o t A = \frac{c o s A}{s i n A} = \frac{\frac{b^{2} + c^{2} - a^{2}}{2 b c}}{a k} = \frac{b^{2} + c^{2} - a^{2}}{2 a b c k}$ $c o t B = \frac{a^{2} + c^{2} - b^{2}}{2 a b c k}$ $c o t C = \frac{a^{2} + b^{2} - c^{2}}{2 a b c k}$ $t o p r o v e 2 c o t B = c o t A + c o t C$ $L H S$ $2 \times \frac{a^{2} + c^{2} - b^{2}}{2 a b c k} = 2 \times \frac{2 b^{2} - b^{2}}{2 a b c k} = \frac{2 b^{2}}{2 a b c k} = \frac{b}{a c k}$ $s i n c e 2 b^{2} = a^{2} + c^{2} a^{2}, b^{2}, c^{2} a r e i n A . P$ $n o w$ $c o t A + c o t C$ $\frac{b^{2} + c^{2} - a^{2}}{2 a b c k} + \frac{a^{2} + b^{2} - c^{2}}{2 a b c k}$ $= \frac{2 b^{2}}{2 a b c k} = \frac{b}{a c k}$ $h e n c e L H S = R H S$ $s o c o t A, c o t B, c o t C a r e i n A P p r o v e d$
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