The value of `sin^(-1)[xsqrt(1-x)-sqrt(x)sqrt(1-x^2)]` is equal to

Options

A. `sin^(-1) x + sin^(-1) sqrtx`
B. `sin^(-1) x - sin^(-1) sqrtx`
C. `sin^(-1) sqrtx - sin^(-1) x`
D. none of these

Explanation

To solve the problem \( \sin^{-1}\left[x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right] \), we will follow these steps:

### Step 1: Substitute Variables
Let \( x = \sin \theta \). Then, \( \sqrt{1 - x} = \sqrt{1 - \sin \theta} = \cos \theta \) and \( \sqrt{1 - x^2} = \sqrt{1 - \sin^2 \theta} = \cos \theta \).

### Step 2: Rewrite the Expression
Now we can rewrite the expression inside the inverse sine:
\[
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