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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