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Differentiate cos^(-1)((1-x^2)/(1+x^2)) ...

Differentiate `cos^(-1)((1-x^2)/(1+x^2))` with respect to `x` , when `x in (0,\ oo)`

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Verified by Experts

Given that,
`y=cos^(-1)((1-x^2)/(1+x^2))`
`x in (0,\ oo)`
`x= tan theta`
so, `y=cos ^-1 [(1-tan^2theta)/(sec^2theta)]`
`=>y=cos^-1(cos^2 theta - sin^2 theta)`
`=>y=cos^-1(cos2theta)`
`x in (0,\ oo)`
`=>tan theta in(0.oo)`
`=>theta in (0, pi/2)`
`y=2theta`
`=>y= 2tan^-1x`
`(dy)/(dx)=2 *d/(dx) tan^-1 x`
`=2/(1+x^2)`
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