If `-1/2 le x le 1/2 then sin^(-1)3x-4x^(3)` equals

Options

A. `3 sin^(-1)x`
B. `pi-3 sin^(-1)x`
C. `-pi -3 sin^(-1)x`
D. none of these

Answer

To solve the problem, we need to evaluate the expression \( \sin^{-1}(3x - 4x^3) \) for \( x \) in the interval \( \left[-\frac{1}{2}, \frac{1}{2}\right] \).

### Step-by-Step Solution:

1. **Identify the Range of x**:
We know that \( x \) is constrained to the interval \( -\frac{1}{2} \leq x \leq \frac{1}{2} \).

**Hint**: Always check the given range of the variable in such problems.

2. **Use the Known Formula**:
There is a known identity for the inverse sine function:
\[ \sin^{-1}(3x - 4x^3) = 3 \sin^{-1}(x) \]
This identity holds true for \( x \) in the interval \( -\frac{1}{2} \leq x \leq \frac{1}{2} \).

**Hint**: Familiarize yourself with common identities involving inverse trigonometric functions.

3. **Apply the Identity**:
Since \( x \) is within the specified range, we can directly apply the identity:
\[ \sin^{-1}(3x - 4x^3) = 3 \sin^{-1}(x) \]

**Hint**: When you recognize a pattern or identity, apply it directly to simplify your calculations.

4. **Final Result**:
Therefore, the expression \( \sin^{-1}(3x - 4x^3) \) simplifies to:
\[ \sin^{-1}(3x - 4x^3) = 3 \sin^{-1}(x) \]

**Hint**: Always ensure to write down the final simplified form clearly.

### Conclusion:
Thus, the answer to the question is:
\[ \sin^{-1}(3x - 4x^3) = 3 \sin^{-1}(x) \]