Solve the equation for x: ^ -1 x + ^ -1 (1 - x) = ^ -1 x - Vidyadip

Question

Solve the equation for $x: \sin^{-1}x + \sin^{-1}(1 - x) = \cos^{-1}x$

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Answer

$\sin^{-1}\text{x} + \sin^{-1}(1 - \text{x}) = \cos^{-1}\text{x}\Rightarrow\sin^{-1}(1 - \text{x})= \frac{\pi}{2} - 2\sin^{-1}\text{x}$
$\Rightarrow1 - \text{x} = \sin\bigg(\frac{\pi}{2}-2\sin^{-1}\text{x}\bigg)\Rightarrow1 - \text{x} = \cos(2\sin^{-1}\text{x})\Rightarrow1 - \text{x} = 1 - 2\sin^{2}(\sin^{-1}\text{x})$
$\Rightarrow1 - \text{x} = 1 - \text{2x}^{2}$
Solving we get, $\text{x} =0 \text{ or x} =\frac{1}{2}$

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