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Question
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
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Solution
\[\text{ Let, y } = \sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}\]
\[\text{ Put, x } = \cos \theta\]
\[ y = \sin^{- 1} \left\{ \sqrt{1 - \cos^2 \theta} \right\}\]
\[ y = \sin^{- 1} \left( \sin\theta \right) . . . \left( 1 \right)\]
\[\text{ Here }, 0 < x < 1\]
\[ \Rightarrow 0 < \cos \theta < 1\]
\[ \Rightarrow 0 < \theta < \frac{\pi}{2}\]
\[\text{ So, from equation} \left( 1 \right), \]
\[ y = \theta \left[ \text{Since }, \sin^{- 1} \left( \sin\theta \right) = \theta, \text{ if } \theta \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \right]\]
\[ y = \cos^{- 1} x \left[ \text{ Since }, x = \cos \theta \right]\]
\[\text{ Differentiating it with respect to x }, \]
\[\frac{d y}{d x} = - \frac{1}{\sqrt{1 - x^2}}\]
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