The values of x satisfying the equation $s i n^{- 1} x + s i n^{- 1} (1 - x) = c o s^{- 1}$ x are :

1, 0

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1, - 1

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0, \frac{1}{2}

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None

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Solution

The correct option is C $0, \frac{1}{2}$
We have, $s i n^{- 1} (1 - x) = (\frac{π}{2} - s i n^{- 1} x) - s i n^{- 1} x$
$= \frac{π}{2} - 2 s i n^{- 1} x$
Taking sine of both sides,
$1 - x = s i n (\frac{π}{2} - 2 s i n^{- 1} x) = c o s (2 s i n^{- 1} x)$
$= c o s^{2} θ = 1 - 2 s i n^{2} θ$ , where $s i n^{- 1} x = θ$
or $1 - x = 1 - 2 x^{2}$ or $x (1 - 2 x) = 0$
$∴$ $x = 0, \frac{1}{2}$

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