Maximizing and minimizing inverse trigo functions

Geometry Level 5

f(x)=(sin1x)3+(cos1x)3\large f(x) = {\left(\sin^{-1}x\right)}^3 + {\left(\cos^{-1}x\right)}^3

For real xx, if the sum of the minimum and maximum values of the function above can be represented as abπ3\dfrac{a}{b} \pi^3, where aa and bb are coprime positive integers, then what is the value of a+ba+b?

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