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Question
Prove that `tan^-1x + tan^-1 (2x)/(1 - x^2) = tan^-1 (3x - x^3)/(1 - 3x^2), |x| < 1/sqrt(3)`
Solution
`tan^-1x + tan ((2x)/(1 - x^2)) = tan^-1 [(x + (2x)/(1 - x^2))/(1 - x((2x)/(1 - x^2)))]`
= `tan^-1 [((x(1 - x^2) + 2x)/(1 - x^2))/((1 - x^2 - 2x^2)/(1 - x^2))]`
= `tan^-1 [(x - x^3 + 2x)/(1 - 3x^2)]`
= `tan^-1 [(3x - x^3)/(1 - 3x^2)]`
If `3x^2 < 1`
⇒ `x^2 < 1/3`
⇒ `|x| < 1/sqrt(3)`
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