The x satisfying $s i n^{- 1} x + s i n^{- 1} (1 - x) = c o s^{- 1} x$ 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 (1-2x)=0
$∴ x = 0, \frac{1}{2}$

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