If `sin (alpha - beta) =1/2 and cos (alpha + beta) =1/2,` where` alpha and beta` are positive acute angles, then `alpha and beta` are

Options

A. `alpha =45^(@) , beta =15^(@)`
B. `alpha =15^(@), beta =45^(@)`
C. `alpha =60^(@) , beta=15^(@)`
D. `alpha = 15^(@), beta =60^(@)`

Answer

To solve the problem, we need to find the values of angles \( \alpha \) and \( \beta \) given that \( \sin(\alpha - \beta) = \frac{1}{2} \) and \( \cos(\alpha + \beta) = \frac{1}{2} \).

### Step-by-step Solution:

1. **Identify the angles for sine and cosine values:**
- We know that \( \sin(30^\circ) = \frac{1}{2} \). Therefore, from the equation \( \sin(\alpha - \beta) = \frac{1}{2} \), we can write:
\[ \alpha - \beta = 30^\circ \quad \text{(Equation 1)} \]

2. **Identify the angle for cosine:**
- We also know that \( \cos(60^\circ) = \frac{1}{2} \). Thus, from the equation \( \cos(\alpha + \beta) = \frac{1}{2} \), we can write:
\[ \alpha + \beta = 60^\circ \quad \text{(Equation 2)} \]

3. **Set up the equations:**
- Now we have two equations:
\[ \alpha - \beta = 30^\circ \quad \text{(1)} \]
\[ \alpha + \beta = 60^\circ \quad \text{(2)} \]

4. **Add the two equations:**
- Adding Equation 1 and Equation 2:
\[ (\alpha - \beta) + (\alpha + \beta) = 30^\circ + 60^\circ \]
\[ 2\alpha = 90^\circ \]
\[ \alpha = 45^\circ \]

5. **Substitute to find \( \beta \):**
- Now substitute \( \alpha = 45^\circ \) back into Equation 1:
\[ 45^\circ - \beta = 30^\circ \]
\[ \beta = 45^\circ - 30^\circ = 15^\circ \]

6. **Final values:**
- Thus, we find:
\[ \alpha = 45^\circ, \quad \beta = 15^\circ \]

### Conclusion:
The values of the angles are \( \alpha = 45^\circ \) and \( \beta = 15^\circ \).