Addition Formulae Question

https://www.quora.com/profile/Bravewarrior/p-144751006
Here is the question, its solution and my working out. Although I got the same answer, can someone please explain the first half of the solution to me? And is that the right way to approach questions like these?

Theyve "centered" the way they use the addition formula so expand it about the center
theta + 2pi/3
and the other two angles are +/-2pi/3 from that one. Its a bit longer than yours, but sometimes centering can make the the series easier to evaulate. Your answer is fine.

They could have noted
cos(theta+4pi/3) = cos(theta-2pi/3)
and written it as
cos(theta) + cos(theta+2pi/3) + cos(theta-2p/3) = ...
Though there isnt much in it for this question.

A trivial way to "answer" the question is to note that it effectively corresponds to walking round the perimeter of a unit length equilateral triangle back to the starting point. So as you return to where you start, the sum/displacement is zero.