Convert the sums or differences into products:
`cos 12 alpha + cos 8 alpha `

Answer

To convert the expression \( \cos 12\alpha + \cos 8\alpha \) into a product, we can use the cosine addition formula. Here are the steps to solve the problem:

### Step-by-step Solution:

1. **Identify the formula**: We will use the formula for the sum of cosines:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]

2. **Assign values**: In our case, let \( A = 12\alpha \) and \( B = 8\alpha \).

3. **Calculate \( A + B \)**:
\[ A + B = 12\alpha + 8\alpha = 20\alpha \]

4. **Calculate \( A - B \)**:
\[ A - B = 12\alpha - 8\alpha = 4\alpha \]

5. **Substitute into the formula**:
\[ \cos 12\alpha + \cos 8\alpha = 2 \cos\left(\frac{20\alpha}{2}\right) \cos\left(\frac{4\alpha}{2}\right) \]

6. **Simplify the expression**:
\[ = 2 \cos(10\alpha) \cos(2\alpha) \]

### Final Answer:
Thus, the expression \( \cos 12\alpha + \cos 8\alpha \) can be written as:
\[ \cos 12\alpha + \cos 8\alpha = 2 \cos(10\alpha) \cos(2\alpha) \]

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