Simplify be reducing to a single term :
`sin 3 alpha cos 2 alpha + cos 3 alpha sin 2 alpha.`

Answer

To simplify the expression \( \sin 3\alpha \cos 2\alpha + \cos 3\alpha \sin 2\alpha \), we can use the sine addition formula. Here’s the step-by-step solution:

### Step 1: Identify the formula
We recognize that the expression \( \sin A \cos B + \cos A \sin B \) can be simplified using the sine addition formula:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B. \]

### Step 2: Assign values to A and B
In our case, we can assign:
- \( A = 3\alpha \)
- \( B = 2\alpha \)

### Step 3: Apply the sine addition formula
Using the sine addition formula, we can rewrite the expression:
\[ \sin 3\alpha \cos 2\alpha + \cos 3\alpha \sin 2\alpha = \sin(3\alpha + 2\alpha). \]

### Step 4: Simplify the argument of sine
Now, we simplify the argument:
\[ 3\alpha + 2\alpha = 5\alpha. \]

### Step 5: Write the final answer
Thus, we can conclude that:
\[ \sin 3\alpha \cos 2\alpha + \cos 3\alpha \sin 2\alpha = \sin 5\alpha. \]

### Final Answer
The simplified expression is:
\[ \sin 5\alpha. \]

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