Differentiate `sin^(-1)((1-x^2)/(1+x^2))+sec^(-1)((1+x^2)/(1-x^2)),\ \ x in R` with respect to `x`

Explanation

Solution:

Let, `y=\sin ^{-1}(\frac{1-x^{2}}{1+x^{2}}\right)+\sec ^{-1}(\frac{1+x^{2}}{1-x^{2}})`

`\Rightarrow y=\sin ^{-1}(\frac{1-x^{2}}{1+x^{2}}\right)+\cos ^{-1}(\frac{1-x^{2}}{1+x^{2}}\right)[Since ^{2} \sec ^{-1} x=\cos ^{-1}(\frac{1}{x}]`

`\Rightarrow y=\frac{\pi}{2} Since , \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}]`

Differentiate it with respect to x,
`\therefore \frac{d y}{d x}=0`