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Question
Write the value of \[\sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right)\]
Solution
We know that
\[\sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2}\] and
`cos^-1(-x)=pi-cos^-1x.`
\[\therefore \sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right) = \sin^{- 1} \left( \frac{1}{3} \right) - \left[ \pi - \cos^{- 1} \left( \frac{1}{3} \right) \right]\]
\[ = \sin^{- 1} \left( \frac{1}{3} \right) - \pi + \cos^{- 1} \left( \frac{1}{3} \right)\]
\[ = \left[ \sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} \left( \frac{1}{3} \right) \right] - \pi\]
\[ = \frac{\pi}{2} - \pi \left[ \because \sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2} \right]\]
\[ = - \frac{\pi}{2}\]
∴ \[\sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right) = - \frac{\pi}{2}\]
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