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Question
Write the value of \[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right), \left| x \right| \leq 1\]
Solution
We have
\[\left| x \right| \leq 1\]
\[ \Rightarrow \pm x \leq 1\]
\[ \Rightarrow x \leq 1 or - x \leq 1\]
\[ \Rightarrow x \leq 1 or x \geq - 1\]
\[ \Rightarrow x \in \left[ - 1, 1 \right]\]
Now,
\[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right) = \cos\left( \frac{\pi}{2} \right) \left[ \because \sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2} \right]\]
\[ = 0\]
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