Differentiate `sin^(-1)((1-x^2)/(1+x^2))`, `0ltxlt1` with respect to `x`

Explanation

Given that,
`y=sin^(-1)((1-x^2)/(1+x^2))`
`0ltxlt1`
`x=tan theta`
so, `y= sin^-1 [(1-tan^2theta)/(1+tan^2theta)]`
`=>y=sin^-1 (cos 2theta)`
`=>y=sin^-1 [sin(pi/2-2theta)]`
`0ltxlt1`
`=>0lt tan theta lt1`
`=>0 lt theta lt pi/4`
`=>0 lt 2 theta lt pi/2`
`=>0 lt pi/2-2theta lt pi/2`
`y= pi/2-2theta`
`=>y=pi/2 -2 tan^-1 x`
`(dy)/(dx)=-2/(1+x^2)`