`sin[2sin^(-1)(4/5)]`

Options

A. `12/25`
B. `16/25`
C. `24/25`
D. none of these

Explanation

To solve the problem \( \sin[2\sin^{-1}(4/5)] \), we will use the formula for the sine of double the inverse sine function.

### Step-by-step Solution:

1. **Identify the formula**: We know that
\[ \sin[2\sin^{-1}(x)] = 2x\sqrt{1 - x^2} \]
In our case, \( x = \frac{4}{5} \).

2. **Substitute \( x \) into the formula**:
\[ \sin[2\sin^{-1}(4/5)] = 2 \cdot \frac{4}{5} \cdot \sqrt{1 - \left(\frac{4}{5}\right)^2} \]

3. **Calculate \( 1 - x^2 \)**:
\[ 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{25 - 16}{25} = \frac{9}{25} \]

4. **Take the square root**:
\[ \sqrt{1 - \left(\frac{4}{5}\right)^2} = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

5. **Substitute back into the sine formula**:
\[ \sin[2\sin^{-1}(4/5)] = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} \]

6. **Calculate the final expression**:
\[ = \frac{24}{25} \]

Thus, the value of \( \sin[2\sin^{-1}(4/5)] \) is \( \frac{24}{25} \).

### Final Answer:
\[ \frac{24}{25} \]