Convert the sums or differences into products:
`sin 12 A - sin 4 A`

Answer

To convert the expression \( \sin 12A - \sin 4A \) into a product, we can use the sine difference identity. The identity states:

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

### Step-by-Step Solution:

1. **Identify C and D**:
- Here, let \( C = 12A \) and \( D = 4A \).

2. **Apply the Identity**:
- Substitute \( C \) and \( D \) into the identity:
\[ \sin 12A - \sin 4A = 2 \cos\left(\frac{12A + 4A}{2}\right) \sin\left(\frac{12A - 4A}{2}\right) \]

3. **Calculate \( C + D \) and \( C - D \)**:
- Calculate \( C + D \):
\[ C + D = 12A + 4A = 16A \]
- Calculate \( C - D \):
\[ C - D = 12A - 4A = 8A \]

4. **Substitute Back into the Expression**:
- Now substitute these results back into the equation:
\[ \sin 12A - \sin 4A = 2 \cos\left(\frac{16A}{2}\right) \sin\left(\frac{8A}{2}\right) \]

5. **Simplify the Fractions**:
- Simplifying the fractions gives:
\[ \sin 12A - \sin 4A = 2 \cos(8A) \sin(4A) \]

### Final Answer:
Thus, the expression \( \sin 12A - \sin 4A \) can be converted into the product:
\[ \sin 12A - \sin 4A = 2 \cos(8A) \sin(4A) \]