If cos `(x-y)=(sqrt3)/2` and `sin(x+y)=1/2`, then the value of `x(0ltxlt90)` is

Options

A. `45^(@)`
B. `30^(@)`
C. `15^(@)`
D. `60^(@)`

Answer

To solve the problem, we need to find the value of \( x \) given that \( \cos(x - y) = \frac{\sqrt{3}}{2} \) and \( \sin(x + y) = \frac{1}{2} \).

### Step-by-step Solution:

1. **Identify the angles from trigonometric values**:
- From \( \cos(x - y) = \frac{\sqrt{3}}{2} \), we know that \( x - y \) can be \( 30^\circ \) or \( 330^\circ \). However, since \( x \) and \( y \) are angles in the range of \( 0^\circ \) to \( 90^\circ \), we will take \( x - y = 30^\circ \).
- From \( \sin(x + y) = \frac{1}{2} \), we know that \( x + y \) can be \( 30^\circ \) or \( 150^\circ \). Again, considering the range of \( x \) and \( y \), we will take \( x + y = 30^\circ \).

2. **Set up the equations**:
- We now have two equations:
\[ x - y = 30^\circ \quad \text{(1)} \]
\[ x + y = 30^\circ \quad \text{(2)} \]

3. **Add the two equations**:
- Adding equations (1) and (2):
\[ (x - y) + (x + y) = 30^\circ + 30^\circ \]
\[ 2x = 60^\circ \]

4. **Solve for \( x \)**:
- Dividing both sides by 2:
\[ x = 30^\circ \]

5. **Conclusion**:
- Thus, the value of \( x \) is \( 30^\circ \).

### Final Answer:
\[ x = 30^\circ \]