I think he doesn't like being called the Tooth Fairy

Above shows a circle inscribed inside a unit square. An arc of a circle was drawn from one vertex of the square to the opposite vertex of the square with its center at the bottom right. If the difference of the two arc lengths of the grey crescent shaded region is given by $\cos^{-1} \left( \sqrt{ \frac AB} \right)$ for coprime positive integers $A$ and $B$ , find the value of $A$ .