sin^(−1) (((2(√x))/(3x)))−sin^(−1) ((√(1−x)) )= sin^(−1) ((1/3))


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Question Number 130423 by john_santu last updated on 25/Jan/21

$\sin^{- 1} (\frac{2 \sqrt{x}}{3 x}) - \sin^{- 1} (\sqrt{1 - x}) = \sin^{- 1} (\frac{1}{3})$
	Answered by liberty last updated on 25/Jan/21

	$l e t p = \sin^{- 1} (\frac{2}{3 \sqrt{x}}); q = \sin^{- 1} (\sqrt{1 - x})$ $w e h a v e \sin (p - q) = \frac{1}{3}$ $\Leftrightarrow \sin p \cos q - \cos p \sin q = \frac{1}{3}$ $\Leftrightarrow \frac{2}{3 \sqrt{x}} . \sqrt{x} - \frac{\sqrt{9 x - 4}}{3 \sqrt{x}} . \sqrt{1 - x} = \frac{1}{3}$ $\Leftrightarrow \sqrt{x} = \sqrt{(9 x - 4) (1 - x)}$ $\Rightarrow x = - 9 x^{2} + 13 x - 4$ $\Rightarrow 9 x^{2} - 12 x + 4 = 0$ $\Rightarrow {(3 x - 2)}^{2} = 0 \Rightarrow x = \frac{2}{3}$
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