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

Answer

To convert the sum \( \sin 12A + \sin 4A \) into a product, we can use the sine addition formula. Here’s a step-by-step solution:

### Step 1: Identify the angles
We have:
- \( c = 12A \)
- \( d = 4A \)

### Step 2: Use the sine addition formula
The formula for the sum of sines is:
\[ \sin c + \sin d = 2 \sin\left(\frac{c + d}{2}\right) \cos\left(\frac{c - d}{2}\right) \]

### Step 3: Calculate \( \frac{c + d}{2} \) and \( \frac{c - d}{2} \)
First, calculate \( c + d \):
\[ c + d = 12A + 4A = 16A \]
Now, calculate \( \frac{c + d}{2} \):
\[ \frac{c + d}{2} = \frac{16A}{2} = 8A \]

Next, calculate \( c - d \):
\[ c - d = 12A - 4A = 8A \]
Now, calculate \( \frac{c - d}{2} \):
\[ \frac{c - d}{2} = \frac{8A}{2} = 4A \]

### Step 4: Substitute back into the formula
Now substitute these values back into the sine addition formula:
\[ \sin 12A + \sin 4A = 2 \sin(8A) \cos(4A) \]

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

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