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Question
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
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Solution
\[\text{Let,} f\left( x \right) = \cos^{- 1} \left( \sin x \right)\]
\[ \Rightarrow f\left( x \right) = \cos^{- 1} \left[ \cos\left( \frac{\pi}{2} - x \right) \right]\]
\[ \Rightarrow f\left( x \right) = \frac{\pi}{2} - x\]
\[\text{ Thus }, f'\left( x \right) = \frac{d}{dx}\left( \frac{\pi}{2} - x \right) = - 1\]
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