Prove that `cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4cos((alpha+beta)/2)cos((beta+gamma)/2)cos((gamma+alpha)/2)`

Explanation

To prove the identity \( \cos \alpha + \cos \beta + \cos \gamma + \cos(\alpha + \beta + \gamma) = 4 \cos\left(\frac{\alpha + \beta}{2}\right) \cos\left(\frac{\beta + \gamma}{2}\right) \cos\left(\frac{\gamma + \alpha}{2}\right) \), we will start from the left-hand side (LHS) and manipulate it to match the right-hand side (RHS).

### Step 1: Start with the Left-Hand Side
We begin with the expression:
\[ \text{LHS} = \cos \alpha + \cos \beta + \cos \gamma + \cos(\alpha + \beta + \gamma) \]

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