Differentiate `tan^-1""((2x)/(1-x^2))` with respect to `cos ^-1 ((1-x^2)/(1+x^2))`

Explanation

Let `y_(1)=tan^-1""(2x)/(1-x^2)=2 tan^-1 x`
`rArr dy_(1)/dx =2 d/dx tan^-1 x=(2)/(1+x^2)`
` Let y_(2)=cos ^-1((1-x^2)/(1+x^2))`
and x tan `theta `
`therefore y_(2)=cos ^-1""(1-tan^2 theta)/(1+tan^2theta)`
`rArr (dy_2)/(dx)=2 d/dx . tan ^-1x=2/(1+x^2)`
`now dy_1/dy_2=(dy_1//dx)/(dy_2//dx)=(2//(1+x^2))/(2//(1+x^2))=1.`