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Question
If 4 cos−1 x + sin−1 x = π, then the value of x is
Options
`2/3`
`1/sqrt2`
`sqrt3/2`
`2/sqrt3`
Solution
(c) `sqrt3/2`
We know that
\[\sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2}\]
\[4 \cos^{- 1} x + \sin^{- 1} x = \pi\]
\[ \Rightarrow 4 \cos^{- 1} x + \frac{\pi}{2} - \cos^{- 1} x = \pi\]
\[ \Rightarrow 3 \cos^{- 1} x = \pi - \frac{\pi}{2}\]
\[ \Rightarrow 3 \cos^{- 1} x = \frac{\pi}{2}\]
\[ \Rightarrow \cos^{- 1} x = \frac{\pi}{6}\]
\[ \Rightarrow x = \cos\frac{\pi}{6}\]
\[ \Rightarrow x = \frac{\sqrt{3}}{2}\]
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