Find the value of
`sin 37 (1/2) .sin7 (1/2)^(@)`

Options

A. `(sqrt(3)-sqrt(2))/(4)`
B. `(sqrt(3)+sqrt(2))/(4)`
C. `(sqrt(3)-sqrt(2))/(2)`
D. None of these

Explanation

To solve the expression \( \sin(37.5^\circ) \cdot \sin(7.5^\circ) \), we can use the trigonometric identity for the product of sines:

\[ 2 \sin A \sin B = \cos(A - B) - \cos(A + B) \]

### Step-by-Step Solution:

1. **Identify \( A \) and \( B \)**:
- Let \( A = 37.5^\circ \) and \( B = 7.5^\circ \).

2. **Apply the identity**:
- According to the identity, we have:
\[ 2 \sin(37.5^\circ) \sin(7.5^\circ) = \cos(37.5^\circ - 7.5^\circ) - \cos(37.5^\circ + 7.5^\circ) \]

3. **Calculate \( A - B \) and \( A + B \)**:
- \( A - B = 37.5^\circ - 7.5^\circ = 30^\circ \)
- \( A + B = 37.5^\circ + 7.5^\circ = 45^\circ \)

4. **Substitute back into the identity**:
\[ 2 \sin(37.5^\circ) \sin(7.5^\circ) = \cos(30^\circ) - \cos(45^\circ) \]

5. **Find the values of \( \cos(30^\circ) \) and \( \cos(45^\circ) \)**:
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
- \( \cos(45^\circ) = \frac{1}{\sqrt{2}} \)

6. **Substitute these values**:
\[ 2 \sin(37.5^\circ) \sin(7.5^\circ) = \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \]

7. **Divide both sides by 2** to find \( \sin(37.5^\circ) \sin(7.5^\circ) \):
\[ \sin(37.5^\circ) \sin(7.5^\circ) = \frac{1}{2} \left( \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \right) \]

8. **Simplify the expression**:
- To combine the terms, find a common denominator:
\[ \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} = \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2\sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{2} \]
- Thus, we have:
\[ \sin(37.5^\circ) \sin(7.5^\circ) = \frac{1}{2} \cdot \frac{\sqrt{3} - \sqrt{2}}{2} = \frac{\sqrt{3} - \sqrt{2}}{4} \]

### Final Answer:
\[ \sin(37.5^\circ) \cdot \sin(7.5^\circ) = \frac{\sqrt{3} - \sqrt{2}}{4} \]