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Question
Show that `2tan^-1x+sin^-1 (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.
Solution
We have
`2tan^-1x+sin^-1 ((2x)/(1+x^2))`
(1) For x > 1,
`=2tan^-1x+sin^-1 ((2x)/(1+x^2))`
`=pi-sin^-1((2x)/(1+x^2))+sin^-1((2x)/(1+x^2))` `[because 2tan^-1x=pi - sin^-1((2x)/(1+x^2)),x>1]`
`=pi`
(2) For x = 1,
`=2tan^-1x+sin^-1 ((2x)/(1+x^2))`
`=2tan^-1(1)+sin^-1((2(1))/(1+(1)^2))`
`=2tan^-1(1)+sin^-1(1)`
`=2(pi/4)+pi/2`
= π
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