Trig and trig inverse integrand!?

Calculus Level 3

\[\int _{ -1 }^{ 1 } \sin { (\arcsin { x }) } \times \sin { (\arccos { x }) } \times \cos { (\arccos { x }) } \times \cos { (\arcsin { x }) } \, dx\]

If the integral above can be expressed in the form \(\dfrac { A }{ B }\), where \({ A }\) and \({ B }\) are positive coprime integers, find \(A+B\).


If you enjoyed the problem, here is part 2.

This problem is original.

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