The number of solution of the equation $1 + x^{2} + 2 x sin ({cos}^{- 1} y) = 0$ is :

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Solution

The correct option is A 1
Given, $1 + x^{2} + 2 x (sin ({cos}^{- 1} (y))) = 0$
$1 + x^{2} + 2 x (sin ({sin}^{- 1} (\sqrt{1 - y^{2}}))) = 0$
$1 + x^{2} + 2 x (\sqrt{1 - y^{2}}) = 0$
$2 x (\sqrt{1 - y^{2}}) = - (1 + x^{2})$
$4 x^{2} (1 - y^{2}) = 1 + x^{4} + 2 x^{2}$
$4 x^{2} - 4 x^{2} y^{2} = 1 + x^{4} + 2 x^{2}$
$- 4 x^{2} (y^{2}) = (1 - x^{2})^{2}$
Hence solution will be $x = 1$ and $y = \frac{1}{2}$

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