91Ó°ÊÓ

To solve the given problem, we start by applying the Angle Sum Formula for cosine. This formula helps us calculate the cosine of the sum of two angles. It's given by:

\(\text{cos}(\theta_1 + \theta_2) = \text{cos}(\theta_1)\text{cos}(\theta_2) - \text{sin}(\theta_1)\text{sin}(\theta_2)\).

For our specific problem, we substitute \( \theta_1 = \text{\alpha} \) and \( \theta_2 = \text{\beta} \):

\(\text{cos}(\text{\alpha + \beta}) = \text{cos \alpha}\text{cos \beta} - \text{sin \alpha}\text{sin \beta}\).

This formula is essential for solving problems involving the combination of two angles. Mastering this identity will help you understand how trigonometric functions interact when angles are added.

Next, we use the Angle Difference Formula for cosine. This formula calculates the cosine of the difference between two angles. The formula is:

\(\text{cos}(\theta_1 - \theta_2) = \text{cos}(\theta_1)\text{cos}(\theta_2) + \text{sin}(\theta_1)\text{sin}(\theta_2)\).

For our problem, substituting \( \theta_1 = \text{\alpha} \) and \( \theta_2 = \text{\beta} \) gives:

\(\text{cos}(\text{\alpha - \beta}) = \text{cos \alpha}\text{cos \beta} + \text{sin \alpha}\text{sin \beta}\).

This formula is the counterpart to the Angle Sum Formula and equally important. It tells us how cosine values operate when angles are subtracted from one another.

Understanding both the sum and difference formulas allows you to manipulate and combine trigonometric functions effectively.

Once we have the expressions for \(\text{cos}(\text{\alpha + \beta})\) and \(\text{cos}(\text{\alpha - \beta})\), we add them together to complete the problem:

\(\text{cos}(\text{\alpha + \beta}) + \text{cos}(\text{\alpha - \beta})\).

When we do this, we have:

\((\text{cos \alpha}\text{cos \beta} - \text{sin \alpha}\text{sin \beta}) + (\text{cos \alpha}\text{cos \beta} + \text{sin \alpha}\text{sin \beta})\).

Notice the terms \(\text{sin \alpha}\text{sin \beta}\) cancel out because one's negative and one's positive. We're left with:

\(\text{cos \alpha}\text{cos \beta} + \text{cos \alpha}\text{cos \beta} = 2 \text{cos \alpha}\text{cos \beta}\).

So, we have established that:

\(\text{cos}(\text{\alpha + \beta}) + \text{cos}(\text{\alpha - \beta}) = 2 \text{cos \alpha}\text{cos \beta}\).

This confirms our original identity and demonstrates the power of trigonometric identities in simplifying complex expressions. By recognizing patterns and using known identities, we can solve intricate problems efficiently.