Prove that:tan−1t+tan−11−t22t=tan−11−3t23t−t3,t being positive, if t 3, and tan−1t+tan−11−t22t=π+tan−11−3t23t−t3 if t>31 and <3

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Prove that: ^{-1} t + ^{-1} {2t}{1 - t^2} = ^{-1} {3t - t^3}{1

Question

Prove that:
$tan^{- 1} t + tan^{- 1} \frac{2 t}{1 - t ^{2}} = tan^{- 1} \frac{3 t - t ^{3}}{1 - 3 t ^{2}}, t$ being positive, if $t < \frac{1}{3}$ or $> 3$ ,
and
$tan^{- 1} t + tan^{- 1} \frac{2 t}{1 - t ^{2}} = π + tan^{- 1} \frac{3 t - t ^{3}}{1 - 3 t ^{2}}$ if $t > \frac{1}{3}$ and $< 3$

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Samuel Discussed

Prove that:

tan^{- 1} t + tan^{- 1} \frac{2 t}{1 - t ^{2}} = tan^{- 1} \frac{3 t - t ^{3}}{1 - 3 t ^{2}}, t

being positive, if

t < \frac{1}{3}

> 3

,
and

tan^{- 1} t + tan^{- 1} \frac{2 t}{1 - t ^{2}} = π + tan^{- 1} \frac{3 t - t ^{3}}{1 - 3 t ^{2}}

t > \frac{1}{3}

and

< 3

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Question Text

Prove that:

tan^{- 1} t + tan^{- 1} \frac{2 t}{1 - t ^{2}} = tan^{- 1} \frac{3 t - t ^{3}}{1 - 3 t ^{2}}, t

being positive, if

t < \frac{1}{3}

> 3

,
and

tan^{- 1} t + tan^{- 1} \frac{2 t}{1 - t ^{2}} = π + tan^{- 1} \frac{3 t - t ^{3}}{1 - 3 t ^{2}}

t > \frac{1}{3}

and

< 3