Q84 / 180 · COMEDK 2018 · Mathematics

$\sin ^{-1}\left[x \sqrt{1-x}-\sqrt{x} \sqrt{1-x^{2}}\right]=$

  • A
    $\sin ^{-1} x-\sin ^{-1} \sqrt{1-x^{2}}$
  • B
    $\sin ^{-1} x+\sin ^{-1} \sqrt{1-x}$
  • Correct answer
    C
    $\sin ^{-1} x-\sin ^{1} \sqrt{x}$
  • D
    $\sin ^{-1} x+\sin ^{-1} \sqrt{x}$

Solution

Let, $y=\sin ^{-1}\left[x \sqrt{1-x}-\sqrt{x} \sqrt{1-x^{2}}\right]$ $$ \begin{aligned} &=\sin ^{-1}\left[x \sqrt{1-(\sqrt{x})^{2}}-\sqrt{x} \sqrt{1-x^{2}}\right] \\ &=\sin ^{-1} x-\sin ^{-1} \sqrt{x} \end{aligned} $$ $\left[\mathrm{Using} \sin ^{-1} x-\sin ^{-1} y=\sin ^{-1}\left[x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right]\right.$

Hence, the correct answer is option: 'C'.

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