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Question
If sin−1 x − cos−1 x = `pi/6` , then x =
Options
`1/2`
`sqrt3/2`
`-1/2`
none of these
Solution
(b) `sqrt3/2`
We know that
\[\sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2}\]
\[\therefore \sin^{- 1} x - \cos^{- 1} x = \frac{\pi}{6}\]
\[ \Rightarrow \frac{\pi}{2} - \cos^{- 1} x - \cos^{- 1} x = \frac{\pi}{6}\]
\[ \Rightarrow - 2 \cos^{- 1} x = \frac{\pi}{6} - \frac{\pi}{2}\]
\[ \Rightarrow - 2 \cos^{- 1} x = - \frac{\pi}{3}\]
\[ \Rightarrow \cos^{- 1} x = \frac{\pi}{6}\]
\[ \Rightarrow x = \cos\frac{\pi}{6}\]
\[ \Rightarrow x = \frac{\sqrt{3}}{2}\]
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