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prove that TANA +TAN(60+A)+TAN(120+A)=3TAN3A
LHS = tanA +[ [tanA+sqrt3] / [1-{(sqrt3)tanA)}] ] +[ [tanA-sqrt3] / [1+[sqrt3}tanA ] ]
ON solving
=> tanA + [8tanA /(1-3sqtanA)]
=> 9tanA-3cubetanA / [1-3sqtanA]
=> 3tan3A = RHS
Hello Krishna !
It is very simple ..
Just expand the function ..
Tan A + (tan 60 + tan A )/(1-tan60.tanA) + (tan (120) + tan A )/(1-tan120.tanA)
tan A + [root(3) + tanA]/[1-root(3).tanA] + [ tan A - root(3) ] /[1 + root(3).tanA]
tanA + [ root(3) + 3.tanA + tanA + root(3).tan^2A + tanA - root(3) -root(3).tan^2A + 3.tanA ] / [ 1 - 3tan^2A ]
tanA + [ 8tanA] / [ 1-3tan^2A ]
[9tanA - 3.tan^3A ]/[1-3tan^2A]
= 3.tan3A ( eqaul to RHS )
Hence proved
Regards
Yagya
askiitians_expert
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