Why is $\cos \theta + \cos (\theta + 2\pi/3) + \cos(\theta + 4\pi/3)=0$?

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Prove that \[\cos(A+B)=\cos A\cos B - \sin A\sin B,\] where the angles \(A\), \(B\) and \(A+B\) are all acute.

By projecting the sides of an equilateral triangle onto a certain line, prove that \[\cos \theta + \cos \left(\theta + \frac{2\pi}{3}\right) + \cos\left(\theta+\frac{4\pi}{3}\right)=0\] and find the value of the expression \[\sin \theta + \sin \left(\theta + \frac{2\pi}{3}\right) + \sin\left(\theta+\frac{4\pi}{3}\right).\]