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Question
If \[\sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} x = \frac{\pi}{2},\] then find x.
Solution
We know that
\[\sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2}\].
We have
\[\sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} x = \frac{\pi}{2}\]
\[ \Rightarrow \sin^{- 1} \left( \frac{1}{3} \right) = \frac{\pi}{2} - \cos^{- 1} x\]
\[ \Rightarrow \sin^{- 1} \left( \frac{1}{3} \right) = \sin^{- 1} x\]
\[ \Rightarrow x = \frac{1}{3}\]
∴ \[x = \frac{1}{3}\]
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