Given $$\alpha$$ + $$\beta$$ + $$\gamma$$ = 2$$\pi$$

$$\Delta = \left| {\matrix{ 1 & {\cos \gamma } & {\cos \beta } \cr {\cos \gamma } & 1 & {\cos \alpha } \cr {\cos \beta } & {\cos \alpha } & 1 \cr } } \right|$$

$$ = 1 - {\cos ^2}\alpha - \cos \gamma (\cos \gamma - \cos \alpha \cos \beta ) + \cos \beta (\cos \alpha \cos \gamma - \cos \beta )$$

$$ = 1 - {\cos ^2}\alpha - {\cos ^2}\beta - {\cos ^2}\gamma + 2\cos \alpha \cos \beta \cos \gamma $$

$$ = {\sin ^2}\alpha - {\cos ^2}\beta - \cos \gamma (\cos \gamma - 2\cos \alpha \cos \beta )$$

$$ = - \cos (\alpha + \beta )\cos (\alpha - \beta ) - \cos \gamma (\cos (2\pi - (\alpha - \beta )) - 2\cos \alpha \cos \beta )$$

$$ = - \cos (2\pi - \gamma )\cos (\alpha - \beta ) - \cos \gamma (\cos (\alpha + \beta ) - 2\cos \alpha \cos \beta )$$

$$ = - \cos \gamma \cos (\alpha - \beta ) + \cos \gamma (\cos \alpha \cos \beta + \sin \alpha \sin \beta )$$

$$ = - \cos \gamma \cos (\alpha - \beta ) + \cos \gamma \cos (\alpha - \beta )$$

$$ = 0$$

So, the system of equation has infinitely many solutions.