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Question
Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?
Solution
\[\text{ Let, y} = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + se c^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right)\]
\[ \Rightarrow y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) \left[ Since, se c^{- 1} x = \cos^{- 1} \left( \frac{1}{x} \right) \right]\]
\[ \Rightarrow y = \frac{\pi}{2} \left[ \text{ Since }, \sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2} \right]\]
Differentiate it with respect to x,
\[\therefore \frac{d y}{d x} = 0\]
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