Find the range of `tan^(-1)((2x)/(1+x^2))`

Explanation

First, we must get the range of
`(2x)/(1+x^(2))=y`
We have `yx^(2)-2x+y=0`
Since x is real, `D ge 0, " i.e., "4-4y^(2) ge 0 " or " -1 le y le 1.` So,
` "tan"^(-1)(y) in [-(pi)/(4),(pi)/(4)]` (As `tan x` is an increasing function)