Convert the sums or differences into products:
`cos 25 ^(@) - cos 37^(@)`

Answer

To convert the expression \( \cos 25^\circ - \cos 37^\circ \) into a product, we will use the cosine difference formula:

\[ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]

### Step-by-Step Solution:

1. **Identify A and B**:
- Let \( A = 25^\circ \) and \( B = 37^\circ \).

2. **Calculate \( A + B \)**:
\[ A + B = 25^\circ + 37^\circ = 62^\circ \]

3. **Calculate \( A - B \)**:
\[ A - B = 25^\circ - 37^\circ = -12^\circ \]

4. **Apply the cosine difference formula**:
\[ \cos 25^\circ - \cos 37^\circ = -2 \sin\left(\frac{62^\circ}{2}\right) \sin\left(\frac{-12^\circ}{2}\right) \]

5. **Simplify the angles**:
- \( \frac{62^\circ}{2} = 31^\circ \)
- \( \frac{-12^\circ}{2} = -6^\circ \)

6. **Substitute back into the formula**:
\[ \cos 25^\circ - \cos 37^\circ = -2 \sin(31^\circ) \sin(-6^\circ) \]

7. **Use the property of sine**:
- Since \( \sin(-\theta) = -\sin(\theta) \):
\[ \sin(-6^\circ) = -\sin(6^\circ) \]

8. **Final expression**:
\[ \cos 25^\circ - \cos 37^\circ = -2 \sin(31^\circ)(-\sin(6^\circ)) = 2 \sin(31^\circ) \sin(6^\circ) \]

### Final Result:
\[ \cos 25^\circ - \cos 37^\circ = 2 \sin(31^\circ) \sin(6^\circ) \]