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\cos(\theta + 60^\circ) + \cos(\theta - 60^\circ) = 2\cos\theta\cos 60^\circ = 2\cos\theta \cdot \frac{1}{2} = \cos\theta
So equation is $\cos\theta = \sqrt{3}$ — impossible. Therefore, the only possibility is the equation was meant to be:
\Rightarrow \cos\theta = 1 \Rightarrow \theta = 0^\circ

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