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prove-tan-1-1-2-tan2A-tan-1-cotA-tan-1-cot-3-A-0-

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Question Number 34500 by Neelam last updated on 07/May/18
prove tan^(−1) ((1/2)tan2A)+tan^(−1) (cotA)+tan^(−1) (cot^3 A)=0
provetan−1(12tan2A)+tan−1(cotA)+tan−1(cot3A)=0
Commented by tanmay.chaudhury50@gmail.com last updated on 07/May/18
let tanA=t tan^(−1) ((1/2)×((2t)/(1−t^2 )))+tan^(−1) ((1/t))+tan^(−1) ((1/t^3 )) =tan^(−1) ((((t/(1−t^2 )) +(1/t))/(1−(1/(1−t^(2 _ ) )))))+tan^(−1) ((1/t^3 )) =tan^(−1) (((1/(t−t^3 ))/((1−t^2 −1)/(1−t^2 ))))+tan^(−1) ((1/t^(3 ) )) tan^(−1) {((1−t^2 )/(t(1−t^2 )(−t^2 )))}+tan^(−1) ((1/t^(3 ) )) =tan^(−1) ((1/(−t^3 )))+tan^(−1) ((1/t^3 )} =−tan((1/t^ ))+tan^(−1) ((1/t^3 ) )=0
lettanA=ttan−1(12×2t1−t2)+tan−1(1t)+tan−1(1t3)=tan−1(t1−t2+1t1−11−t2)+tan−1(1t3)=tan−1(1t−t31−t2−11−t2)+tan−1(1t3)tan−1{1−t2t(1−t2)(−t2)}+tan−1(1t3)=tan−1(1−t3)+tan−1(1t3}=−tan(1t)+tan−1(1t3)=0

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