# 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 $$\frac { A\cdot B }{ C\cdot D }$$.

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