What is sin`[sin^(-1)((3)/(5))+sin^(-1)((4)/(5))]` equal to ?

Options

A. 0
B. `1//2`
C. 1
D. 2

Explanation

To solve the problem \( \sin\left[\sin^{-1}\left(\frac{3}{5}\right) + \sin^{-1}\left(\frac{4}{5}\right)\right] \), we can use the formula for the sine of the sum of two inverse sine functions:

\[ \sin\left[\sin^{-1}(x) + \sin^{-1}(y)\right] = x\sqrt{1 - y^2} + y\sqrt{1 - x^2} \]

### Step-by-Step Solution:

1. **Identify the values of \( x \) and \( y \)**:
- Let \( x = \frac{3}{5} \) and \( y = \frac{4}{5} \).

2. **Calculate \( \sqrt{1 - y^2} \)**:
- \( y^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \)
- Therefore, \( 1 - y^2 = 1 - \frac{16}{25} = \frac{25 - 16}{25} = \frac{9}{25} \)
- Thus, \( \sqrt{1 - y^2} = \sqrt{\frac{9}{25}} = \frac{3}{5} \).

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

4. **Substitute into the formula**:
- Now we can substitute \( x \), \( y \), \( \sqrt{1 - y^2} \), and \( \sqrt{1 - x^2} \) into the formula:
\[ \sin\left[\sin^{-1}\left(\frac{3}{5}\right) + \sin^{-1}\left(\frac{4}{5}\right)\right] = \frac{3}{5} \cdot \frac{3}{5} + \frac{4}{5} \cdot \frac{4}{5} \]

5. **Calculate the expression**:
\[ = \frac{9}{25} + \frac{16}{25} = \frac{9 + 16}{25} = \frac{25}{25} = 1 \]

6. **Final Result**:
\[ \sin\left[\sin^{-1}\left(\frac{3}{5}\right) + \sin^{-1}\left(\frac{4}{5}\right)\right] = 1 \]

### Conclusion:
The final answer is \( 1 \).