Prove that `2 sin^2 beta + 4 cos(alpha + beta) sin alpha sin beta + cos 2(alpha + beta) = cos 2alpha`

Explanation

To prove that \( 2 \sin^2 \beta + 4 \cos(\alpha + \beta) \sin \alpha \sin \beta + \cos 2(\alpha + \beta) = \cos 2\alpha \), we will start with the left-hand side (LHS) and simplify it step by step.

### Step-by-Step Solution:

1. **Write down the LHS:**
\[ \text{LHS} = 2 \sin^2 \beta + 4 \cos(\alpha + \beta) \sin \alpha \sin \beta + \cos 2(\alpha + \beta) \]
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