(i) Prove that \(\cos(\theta - 60^\circ) + \cos(\theta + 60^\circ) \equiv \cos \theta\).

(ii) Given that \(\frac{\cos(2x - 60^\circ) + \cos(2x + 60^\circ)}{\cos(x - 60^\circ) + \cos(x + 60^\circ)} = 3\), find the exact value of \(\cos x\).