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

\[\text{ Let, y} = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + se c^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right)\]

\[ \Rightarrow y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) \left[ Since, se c^{- 1} x = \cos^{- 1} \left( \frac{1}{x} \right) \right]\]

\[ \Rightarrow y = \frac{\pi}{2} \left[ \text{ Since }, \sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2} \right]\]

Differentiate it with respect to x,

\[\therefore \frac{d y}{d x} = 0\]