<s>[INST] $\alpha + \beta + \gamma = \pi$ , show that $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 2\cos\alpha \cos\beta \cos\gamma = 1$ 
$\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 2\cos\alpha \cos\beta \cos\gamma = 1$

I really didn't know how to solve this problem and I am very unused to the utilization of trigonometric identities, I was wondering if I may have some assistance in this problem with detailed explanations
I was thinking more along the lines of making all of the angles in terms of one angle
 [/INST] I think you mean the following problem.

Let $\alpha+\beta+\gamma=\pi$. Prove that:
  $$\cos^2\alpha+\cos^2\beta+\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma=1$$

We need to prove that
$$\cos^2\alpha+\cos^2\beta-2\cos\alpha\cos\beta\cos(\alpha+\beta)=\sin^2(\alpha+\beta)$$ or
$$\cos^2\alpha+\cos^2\beta-2\cos\alpha\cos\beta(\cos\alpha\cos\beta-\sin\alpha\sin\beta)=(\sin\alpha\cos\beta+\cos\alpha\sin\beta)^2$$ or
$$\cos^2\alpha+\cos^2\beta-2\cos^2\alpha\cos^2\beta=\sin^2\alpha\cos^2\beta+\cos^2\alpha\sin^2\beta$$ or
$$\cos^2\alpha(1-\sin^2\beta)+\cos^2\beta(1-\sin^2\alpha)-2\cos^2\alpha\cos^2\beta=0,$$
which is obvious.
Done!
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