If `f(x)=sin^(-1)((sqrt(3))/2x-1/2sqrt(1-x^2)),-1/2lt=xlt=1,t h e nf(x)` is equal to `sin^(-1)(1/2)-sin^(-1)(x)` `sin^(-1)x-pi/6` `sin^(-1)x+pi/6` (d) none of these

Options

A. `sin^(-1) ((1)/(2)) - sin^(-1) (x)`
B. `sin^(-1) x - (pi)/(6)`
C. `sin^(-1) x + (pi)/(6)`
D. none of these

Explanation

To solve the problem, we need to find \( f(x) \) given by the expression:

\[ f(x) = \sin^{-1}\left(\frac{\sqrt{3}}{2}x - \frac{1}{2}\sqrt{1-x^2}\right) \]

where \( -\frac{1}{2} < x < 1 \).

### Step 1: Substitute \( x \) with \( \sin \theta \)

Let’s assume \( x = \sin \theta \). Then we can rewrite \( f(x) \) as:

\[ f(\sin \theta) = \sin^{-1}\left(\frac{\sqrt{3}}{2}\sin \theta - \frac{1}{2}\sqrt{1 - \sin^2 \theta}\right) \]

### Step 2: Simplify the expression

Using the identity \( \sqrt{1 - \sin^2 \theta} = \cos \theta \), we can substitute this into our expression:

\[ f(\sin \theta) = \sin^{-1}\left(\frac{\sqrt{3}}{2}\sin \theta - \frac{1}{2}\cos \theta\right) \]

### Step 3: Recognize the sine subtraction formula

The expression inside the sine inverse can be recognized as:

\[ \frac{\sqrt{3}}{2}\sin \theta - \frac{1}{2}\cos \theta = \sin\left(\theta - \frac{\pi}{6}\right) \]

This is due to the sine subtraction formula:

\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]

where \( a = \theta \) and \( b = \frac{\pi}{6} \).

### Step 4: Rewrite the function

Thus, we can rewrite \( f(\sin \theta) \) as:

\[ f(\sin \theta) = \sin^{-1}\left(\sin\left(\theta - \frac{\pi}{6}\right)\right) \]

### Step 5: Simplify using the properties of the sine inverse

Since \( \sin^{-1}(\sin x) = x \) for \( x \) in the range \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \), we have:

\[ f(\sin \theta) = \theta - \frac{\pi}{6} \]

### Step 6: Substitute back for \( \theta \)

Recall that we set \( x = \sin \theta \), which implies \( \theta = \sin^{-1}(x) \). Therefore, we can substitute back:

\[ f(x) = \sin^{-1}(x) - \frac{\pi}{6} \]

### Conclusion

Thus, the final result is:

\[ f(x) = \sin^{-1}(x) - \frac{\pi}{6} \]

### Answer

The correct option is:

\[ \text{(b) } \sin^{-1}x - \frac{\pi}{6} \]