A mere square

Geometry Level 5

The following shown here is a portion of the graph for \(f\left( x \right) ={ \left( \arcsin { \cos { x } } \right) }^{ 2 }\). In each region bound by the \(x\)-axis and \(f\left( x \right)\), a square can be drawn, with its top two vertices touching \(f\left( x \right)\), and its bottom two vertices touching the \(x\)-axis, as pictured.

Considering the periodic properties of \(f\left( x \right)\), the area of any of these squares can be expressed in the form \[{ \pi }^{ A }+B \pi +B -C(\pi + A )\sqrt {\pi + D }\] with \({ A }\) a prime number, \({ B }\) a perfect cube, and \({ C }\) and \({ D }\) perfect squares. Find \(\frac { A\cdot B }{ C\cdot D }\).


This problem is original. The picture of the graph was produced from Desmos and the orange square was added with Microsoft Paint.

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