1. Datos del problema:
Suma de cosenos con ángulos en progresión aritmética.

2. Fórmulas usadas:

• Identidad de suma de cosenos: $\sum_{k=1}^n \cos(k\alpha) = \frac{\sin(n\alpha/2) \cos((n+1)\alpha/2)}{\sin(\alpha/2)}$

3. Desarrollo paso a paso:
Sea $S = \cos \frac{2\pi}{7} + \cos \frac{4\pi}{7} + \cos \frac{6\pi}{7}$. Multiplicamos por $2 \sin \frac{\pi}{7}$:
$$ \begin{aligned} 2 \sin \frac{\pi}{7} S &= 2 \sin \frac{\pi}{7} \cos \frac{2\pi}{7} + 2 \sin \frac{\pi}{7} \cos \frac{4\pi}{7} + 2 \sin \frac{\pi}{7} \cos \frac{6\pi}{7} \\ &= (\sin \frac{3\pi}{7} - \sin \frac{\pi}{7}) + (\sin \frac{5\pi}{7} - \sin \frac{3\pi}{7}) + (\sin \frac{7\pi}{7} - \sin \frac{5\pi}{7}) \\ &= \sin \pi - \sin \frac{\pi}{7} \\ &= 0 - \sin \frac{\pi}{7} = -\sin \frac{\pi}{7} \end{aligned} $$
Despejando $S$:
$$ S = \frac{-\sin \frac{\pi}{7}}{2 \sin \frac{\pi}{7}} = -\frac{1}{2} $$

4. Conclusión:
$$ \boxed{-\frac{1}{2} = -\frac{1}{2}} $$