Prove the trigonometric identity manually & by using MATLAB

Prove that:

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

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