Certainly! Let's verify the trigonometric equation [tex]\(\cos(\alpha + \beta) + \cos(\alpha - \beta) = 2 \cos \alpha \cos \beta\)[/tex] step-by-step.
### Step-by-Step Solution
1. Recall the trigonometric sum and difference identities:
- [tex]\(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\)[/tex]
- [tex]\(\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\)[/tex]
2. Express [tex]\(\cos(\alpha + \beta)\)[/tex] using the sum identity:
[tex]\[
\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta
\][/tex]
3. Express [tex]\(\cos(\alpha - \beta)\)[/tex] using the difference identity:
[tex]\[
\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
\][/tex]
4. Add the two identities:
[tex]\[
\cos(\alpha + \beta) + \cos(\alpha - \beta) = (\cos \alpha \cos \beta - \sin \alpha \sin \beta) + (\cos \alpha \cos \beta + \sin \alpha \sin \beta)
\][/tex]
5. Simplify the expression:
- Combine like terms:
[tex]\[
(\cos \alpha \cos \beta - \sin \alpha \sin \beta) + (\cos \alpha \cos \beta + \sin \alpha \sin \beta)
\][/tex]
- The [tex]\(\sin \alpha \sin \beta\)[/tex] terms cancel each other:
[tex]\[
\cos \alpha \cos \beta + \cos \alpha \cos \beta
\][/tex]
6. Factor out the common term:
[tex]\[
\cos(\alpha + \beta) + \cos(\alpha - \beta) = 2 \cos \alpha \cos \beta
\][/tex]
Thus, we have shown that the given trigonometric equation:
[tex]\[
\cos(\alpha + \beta) + \cos(\alpha - \beta) = 2 \cos \alpha \cos \beta
\][/tex]
is indeed true.
Therefore, the equation [tex]\(\cos(\alpha + \beta) + \cos(\alpha - \beta) = 2 \cos \alpha \cos \beta\)[/tex] is a trigonometric identity and holds for all values of [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex].
