Question

Differentiate

${\sin }^{-1}\left(\frac{1-x^2}{1+x^2}\right)+{\sec }^{-1}\left(\frac{1+x^2}{1-x^2}\right),x\in R$

Answer 1

$$\begin{gathered} \mathrm{Let},y={\sin }^{-1}\left(\frac{1-x^2}{1+x^2}\right)+se{c}^{-1}\left(\frac{1+x^2}{1-x^2}\right) \\ \Rightarrow y={\sin }^{-1}\left(\frac{1-x^2}{1+x^2}\right)+{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)\left[\mathrm{Since},se{c}^{-1}x={\cos }^{-1}\left(\frac{1}{x}\right)\right] \\ \Rightarrow y=\frac{\mathrm{\pi }}{2}\left[\mathrm{Since},{\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2}\right] \end{gathered}$$

Differentiate it with respect to x,

$\therefore \frac{dy}{dx}=0$