Prove the trigonometric identity manually & by using MATLAB

Prove that:

\(A\cos \left(\omega t\right) + A\cos \left(\omega t+120^{\circ}\right)+ A\cos \left(\omega t+240^{\circ}\right) = 0\)

Taking left hand side and expand the sum using trigonometric identity, we get: $$=A\cos \left(\omega t\right) + A\left(-\frac{1}{2}\cos \left(\omega t\right)-\frac{\sqrt{3}}{2}\sin \left(\omega t\right)\right) + A\left(-\frac{1}{2}\cos \left(\omega t\right)+\frac{\sqrt{3}}{2}\sin \left(\omega t\right)\right)$$ where, $$\cos \left(\alpha +\beta \right) = \cos \left(\alpha \right)\cos \left(\beta \right) +\sin \left(\alpha \right)\sin \left(\beta \right)$$ and $$\cos\left(120^{\circ} \right) = -\frac{1}{2}, \qquad \sin \left(120^{\circ} \right) = \frac{\sqrt{3}}{2}$$ $$\cos\left(240^{\circ} \right) = -\frac{1}{2}, \qquad \sin \left(240^{\circ} \right) = -\frac{\sqrt{3}}{2}$$ Simplifying the terms by adding/subtracting the coefficients we get: $$=A\cos \left(\omega t\right) -\frac{1}{2}A \cos \left(\omega t\right)-\frac{\sqrt{3}}{2}A \sin \left(\omega t\right) + -\frac{1}{2} A \cos \left(\omega t\right)+\frac{\sqrt{3}}{2} A \sin \left(\omega t\right)$$ $$=A\cos \left(\omega t\right) - A \cos \left(\omega t\right)$$ $$= 0$$

MATLAB Command to execute the Problem

The above problem can be solved using "expand" and "cosd"; cosine function in degress, built-in MATLAB commands to get the solution. Assume the variables as A, w \((\omega)\) and t.

syms A w t
Sol = expand(A*cosd(w*t) + A*cosd(w*t + 120) + A*cosd(w*t + 240))

Sol =
 
0