Question

Show that `2tan^-1x+sin^-1  (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.

Answer 1

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