Simplify be reducing to a single term :
`cos (theta + alpha ) cos ( theta - alpha ) - sin ( theta + alpha ) sin (theta - alpha). `

Answer

To simplify the expression \( \cos(\theta + \alpha) \cos(\theta - \alpha) - \sin(\theta + \alpha) \sin(\theta - \alpha) \), we can use the cosine addition formula.

### Step-by-Step Solution:

1. **Identify the Expression**:
We start with the expression:
\[ \cos(\theta + \alpha) \cos(\theta - \alpha) - \sin(\theta + \alpha) \sin(\theta - \alpha) \]

2. **Apply the Cosine Addition Formula**:
Recall the cosine addition formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
Here, let \( A = \theta + \alpha \) and \( B = \theta - \alpha \). Thus, we can rewrite the expression as:
\[ \cos((\theta + \alpha) + (\theta - \alpha)) \]

3. **Simplify the Argument**:
Now, simplify the argument:
\[ (\theta + \alpha) + (\theta - \alpha) = \theta + \alpha + \theta - \alpha = 2\theta \]
Therefore, the expression simplifies to:
\[ \cos(2\theta) \]

4. **Final Result**:
Thus, the simplified form of the given expression is:
\[ \cos(2\theta) \]