1. Valores y propiedades:

• $\alpha = \arccos \frac{1}{2} = \frac{\pi}{3} \implies \cos \alpha = \frac{1}{2}, \text{sen } \alpha = \frac{\sqrt{3}}{2}$.


• Sea $\beta = \arccos \left( -\frac{1}{7} \right) \implies \cos \beta = -\frac{1}{7}$.


• $\text{sen } \beta = \sqrt{1 - (-\frac{1}{7})^2} = \sqrt{\frac{48}{49}} = \frac{4\sqrt{3}}{7}$.

2. Desarrollo paso a paso:
Aplicamos el coseno de la suma $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \text{sen } \alpha \text{sen } \beta$:
$$\cos(\alpha + \beta) = \left(\frac{1}{2}\right)\left(-\frac{1}{7}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{4\sqrt{3}}{7}\right)$$
$$\cos(\alpha + \beta) = -\frac{1}{14} - \frac{12}{14} = -\frac{13}{14}$$

3. Resultado final:
$\alpha + \beta = \arccos \left( -\frac{13}{14} \right)$. Queda demostrada la identidad.