sin-1-2-x-3x-sin-1-1-x-sin-1-1-3-

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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