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

$p r o v e$ $t {an}^{- 1} (\frac{1}{2} t a n 2 A) + \tan^{- 1} (c o t A) + \tan^{- 1} ({c o t}^{3} A) = 0$
Commented by tanmay.chaudhury50@gmail.com last updated on 07/May/18
$let tanA=t tan^(−1) ((1/2)×((2t)/(1−t^2 )))+tan^(−1) ((1/t))+tan^(−1) ((1/t^3 )) =tan^(−1) ((((t/(1−t^2 )) +(1/t))/(1−(1/(1−t^(2 _ ) )))))+tan^(−1) ((1/t^3 )) =tan^(−1) (((1/(t−t^3 ))/((1−t^2 −1)/(1−t^2 ))))+tan^(−1) ((1/t^(3 ) )) tan^(−1) {((1−t^2 )/(t(1−t^2 )(−t^2 )))}+tan^(−1) ((1/t^(3 ) )) =tan^(−1) ((1/(−t^3 )))+tan^(−1) ((1/t^3 )} =−tan((1/t^ ))+tan^(−1) ((1/t^3 ) )=0$
$l e t t a n A = t$ ${t a n}^{- 1} (\frac{1}{2} \times \frac{2 t}{1 - t^{2}}) + {t a n}^{- 1} (\frac{1}{t}) + {t a n}^{- 1} (\frac{1}{t^{3}})$ $= {t a n}^{- 1} (\frac{\frac{t}{1 - t^{2}} + \frac{1}{t}}{1 - \frac{1}{1 - t^{2_{}}}}) + {t a n}^{- 1} (\frac{1}{t^{3}})$ $= {t a n}^{- 1} (\frac{\frac{1}{t - t^{3}}}{\frac{1 - t^{2} - 1}{1 - t^{2}}}) + {t a n}^{- 1} (\frac{1}{t^{3}})$ ${t a n}^{- 1} {\frac{1 - t^{2}}{t (1 - t^{2}) (- t^{2})}} + {t a n}^{- 1} (\frac{1}{t^{3}})$ $= {t a n}^{- 1} (\frac{1}{- t^{3}}) + {t a n}^{- 1} (\frac{1}{t^{3}}}$ $= - t a n (\frac{1}{t^{}}) + {t a n}^{- 1} (\frac{1}{t^{3}}) = 0$
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