If cos `alpha + cos beta + cos gamma = 0` and also `sin alpha + sin beta + sin gamma= 0,` then prove that: ` cos 3 alpha + cos 3beta + cos 3gamma = 3 cos (alpha + beta + gamma)`

Explanation

To prove that \( \cos 3\alpha + \cos 3\beta + \cos 3\gamma = 3 \cos(\alpha + \beta + \gamma) \) given that \( \cos \alpha + \cos \beta + \cos \gamma = 0 \) and \( \sin \alpha + \sin \beta + \sin \gamma = 0 \), we can follow these steps:

### Step 1: Define Complex Numbers
Let:
\[ a = e^{i\alpha}, \quad b = e^{i\beta}, \quad c = e^{i\gamma} \]
This means:
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