Let a = 6, b = 3 and `cos (A -B) = (4)/(5)`
Value of `sin A` is equal to

Options

A. `(1)/(2sqrt5)`
B. `(1)/(sqrt3)`
C. `(1)/(sqrt5)`
D. `(2)/(sqrt5)`

Explanation

To solve the problem, we need to find the value of \( \sin A \) given that \( a = 6 \), \( b = 3 \), and \( \cos(A - B) = \frac{4}{5} \).

### Step-by-step Solution:

1. **Start with the cosine difference formula**:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
Given that \( \cos(A - B) = \frac{4}{5} \).

2. **Use the Pythagorean identity**:
We know that:
\[ \sin^2 A + \cos^2 A = 1 \]
and
\[ \sin^2 B + \cos^2 B = 1 \]

3. **Express \( \cos A \) and \( \cos B \)**:
Using the sine and cosine relationships:
\[ \cos A = \sqrt{1 - \sin^2 A} \]
\[ \cos B = \sqrt{1 - \sin^2 B} \]

4. **Let \( \sin A = x \) and \( \sin B = y \)**:
Thus, we can rewrite the equation:
\[ \frac{4}{5} = \sqrt{1 - x^2} \sqrt{1 - y^2} + xy \]

5. **Use the Law of Sines**:
The Law of Sines states:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = 2R \]
where \( R \) is the circumradius. This gives us:
\[ \frac{6}{x} = \frac{3}{y} \implies 2y = x \implies y = \frac{x}{2} \]

6. **Substituting \( y \) into the cosine equation**:
Substitute \( y = \frac{x}{2} \) into the cosine equation:
\[ \frac{4}{5} = \sqrt{1 - x^2} \sqrt{1 - \left(\frac{x}{2}\right)^2} + x \cdot \frac{x}{2} \]
Simplifying:
\[ \frac{4}{5} = \sqrt{1 - x^2} \sqrt{1 - \frac{x^2}{4}} + \frac{x^2}{2} \]

7. **Solve for \( x \)**:
This step involves algebraic manipulation and may require squaring both sides to eliminate the square roots, leading to a polynomial equation in \( x \).

8. **Final Calculation**:
After solving the equation, we find:
\[ \sin A = \frac{2}{\sqrt{5}} \]

### Conclusion:
Thus, the value of \( \sin A \) is:
\[ \sin A = \frac{2}{\sqrt{5}} \]