If `y=sin^(-1)((2x)/(1+x^2))+sec^(-1)((1+x^2)/(1-x^2)),0ltxlt1,` prove that`(dy)/ (dx)="4/(1+x^2)`

Explanation

` y = sin^-1((2x)/(1+x^2)) + sec^-1((1+x^2)/(1-x^2))`

` => y = sin^-1((2x)/(1+x^2)) + cos^-1((1-x^2)/(1+x^2))`

Let `x = tan theta`

Then, ` y = sin^-1((2tan theta)/(1+tan^2 theta)) + cos^-1((1-tan^2 theta)/(1+tan^2 theta))`

`=> y = sin^-1(sin 2theta) + cos^-1(cos2 theta))`

Now, `0 lt x lt 1 => 0 lt tan theta lt 1 => 0 lt 2theta lt pi/2`

So, `2theta` is in range of `sin^-1 and cos^-1.`

`:. y = 2theta+2theta`

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