Real Part

Let $(\cos { 60° } +i\sin { 60° } )=w$ . Then, we can rewrite the expression as

${ w }^{ 12 }+{ w }^{ 11 }+...+w+1+11=\frac { { w }^{ 13 }-1 }{ w-1 } +11 =\frac { { (\cos { 60° } +i\sin { 60° } ) }^{ 13 }-1 }{ (\cos { 60° } +i\sin { 60° } )-1 } +11$

Applying De Moivre's Formula,

${ (\cos { 60° } +i\sin { 60° } ) }^{ 13 }=\cos { (13\cdot 60°) } +i\sin { (13\cdot 60°) } =\cos { 780° } +i\sin { 780° } =\cos { 60° } +i\sin { 60° }$

Therefore the equation is simply

$\frac { (\cos { 60° } +i\sin { 60° } )-1 }{ (\cos { 60° } +i\sin { 60° } )-1 } +11=12$

We note that $\cos 60^\circ + i \sin 60^\circ = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} = e^\frac{\pi}{3} = \omega$ is the $6^{th}$ root of unit. Then, we have $\omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 1 = 0$ , $\omega^6 = 1$ and $\omega^n = \omega^{n \text{ mod } 6}$ . Therefore,

$\begin{aligned} S & = (\cos 60^\circ + i \sin 60^\circ)^{12} + (\cos 60^\circ + i \sin 60^\circ)^{11} + ... + (\cos 60^\circ + i \sin 60^\circ) + 12 \\ & = \omega^{12} + \omega^{11} + \omega^{10} + \omega^9 + \omega^8 + \omega^7 + \omega^6 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega + 12 \\ & = (1 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega) + (1 + \omega^5 + \omega^4 + \omega^3 + \omega^2 + \omega) + 12 \\ & = 0 + 0 + 12 \\ & = \boxed{12} \end{aligned}$

Real parts will be simply cos(60) + cos(120) + cos(180) +... cos(720) + 12 (Using De Moivre's Theorem)

For evaluating this series we have

When the angles inside sine or cosine are in AP then we have these set of formulae which are very well known

http://evergreen.loyola.edu/mpknapp/www/papers/knapp-sv.pdf

If we substituted these values in expression we would get value as 0

Hence real part = 12