Problem 40 - Precalculus Essentials

Question

Verify each identity. $$\cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta$$

Step-by-Step Solution

Answer

The left side of given identity, when applied the cosine addition and subtraction identities and simplified, matches exactly with the right side. Hence, the given identity \( \cos (\alpha+\beta)+\cos (\alpha-\beta)=2 \cos \alpha \cos \beta \) has been verified.

Step 1: Apply the cosine addition and subtraction identities

To begin with, the given expression is: \(\cos (\alpha+\beta)+\cos (\alpha-\beta)\) and we need to transform it into the form: \(2 \cos \alpha \cos \beta\).Since the Cosine of the sum of two angles identity is \( \cos (\alpha+\beta)= \cos \alpha \cos \beta - \sin \alpha \sin \beta\) and the Cosine of the difference of two angles identity is \(\cos (\alpha-\beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\), we can write the original expression as:\(\cos (\alpha+\beta)+\cos (\alpha-\beta)= \cos \alpha \cos \beta - \sin \alpha \sin \beta + \cos \alpha \cos \beta + \sin \alpha \sin \beta\)

Step 2: Simplify the expression

Adding the similar terms from the right side, it leads to: \( 2\cos \alpha \cos \beta\). Thus simplifying the given identity now leads to the desired form.