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 given problem, we need to evaluate the expression \( \sin^{-1}\left[x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right] \).

### Step 1: Identify the expression
We start with the expression:
\[ \sin^{-1}\left[x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right] \]

### Step 2: Use the property of inverse trigonometric functions
We can use the property of inverse trigonometric functions:
\[ \sin^{-1}(x) - \sin^{-1}(y) = \sin^{-1\left[x\sqrt{1-y^2} - y\sqrt{1-x^2}\right]} \]
Here, we need to identify \(x\) and \(y\) from our expression.

### Step 3: Identify \(x\) and \(y\)
In our case, we can let:
- \(x = x\)
- \(y = \sqrt{x}\)

### Step 4: Substitute into the property
Now substituting into the property:
\[ \sin^{-1}(x) - \sin^{-1}(\sqrt{x}) = \sin^{-1}\left[x\sqrt{1 - (\sqrt{x})^2} - \sqrt{x}\sqrt{1 - x^2}\right] \]
This simplifies to:
\[ \sin^{-1}(x) - \sin^{-1}(\sqrt{x}) = \sin^{-1}\left[x\sqrt{1 - x} - \sqrt{x}\sqrt{1 - x^2}\right] \]

### Step 5: Conclude the result
Thus, we can conclude that:
\[ \sin^{-1}\left[x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right] = \sin^{-1}(x) - \sin^{-1}(\sqrt{x}) \]

### Final Answer
The value of \( \sin^{-1}\left[x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right] \) is equal to:
\[ \sin^{-1}(x) - \sin^{-1}(\sqrt{x}) \]