$$ arccos \frac{1}{2} = \frac{\pi}{3} $$, $$ arccos(-\frac{1}{2}) = \frac{2\pi}{3} $$. Следовательно, $$ arccos \frac{1}{2} - arccos(-\frac{1}{2}) = \frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3} $$.
Ответ: $$ -\frac{\pi}{3} $$