Question

Find the value of : ${\sin }^{-1}[x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^2}]$

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

Answer 1

The correct option is C ${\sin }^{-1}x-{\sin }^{-1}\sqrt{x}$

We know that: $si{n}^{-1}x\pm si{n}^{-1}y=si{n}^{-1}[x\sqrt{1-y^2}\pm y\sqrt{1-x^2}]$

Comparing with the above equation,
$si{n}^{-1}[x\sqrt{1-{\sqrt{x}}^2}-\sqrt{x}\sqrt{1-x^2}]$
$=si{n}^{-1}\left(x\right)-si{n}^{-1}\left(\sqrt{x}\right)$