Complex or Simple

$\begin{aligned} \frac ab + \frac bc + \frac ca & = 1 \\ \frac {e^{i\alpha}}{e^{i\beta}} + \frac {e^{i\beta}}{e^{i\gamma}} + \frac {e^{i\gamma}}{e^{i\alpha}} & = 1 \\ e^{i(\alpha - \beta)} + e^{i(\beta - \gamma)} + e^{i(\gamma - \alpha)} & = 1 & \small \color{#3D99F6} \text{Using Euler's formula: }e^{i\theta} = \cos \theta + i\sin \theta \\ \cos (\alpha - \beta) + i\sin (\alpha - \beta) + \cos (\beta - \gamma) + i \sin (\beta - \gamma) + \cos (\gamma - \alpha) + i\sin (\gamma - \alpha) & = 1 \\ \sum_{cyc} \cos (\alpha - \beta) + i \sum_{cyc} \sin (\alpha - \beta) & = 1 + i0 & \small \color{#3D99F6} \text{Equating the real part} \\ \implies \sum_{cyc} \cos (\alpha - \beta) & = \boxed{1} \end{aligned}$