If `y = 2tan^(-1)x+sin^(-1)((2x)/(1+x^(2)))`, then `"………"lt y lt "………"`.

Answer

N/a

Explanation

We have, `y = 2tan^(-1)x + sin^(-1)'(2x)/(1+x^(2))`
`:. Y = 2 tan^(-1) tan theta + sin^(-1)'(2tantheta)/(1+tan^(2)theta)` , [let `x = tantheta`]
`rArr y = 2 theta + sin^(-1)sin2 theta` , `[:' sin 2 theta = (2tantheta)/(1+tan^(-2)theta)]`
`rArr y = 2 theta + 2 theta = 4 theta , [:' theta = tan^(-1)x]`
`rArr y = 4 tan^(-1)x`
`:' -pi//2 lt tan^(-1) x lt pi//2`
`:. - (4pi)/(2) lt 4 tan^(-1) x lt 4pi//2`
`rArr -2pi lt 4 tan^(-1) x lt 2x` ltbr gt `rArr -2 pi lt y lt 2pi , [:' y = tan^(-1) x]`