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Geometry Level 3

If cot1(1+sinx+1sinx1+sinx1sinx)=Ax\cot^{-1} \left(\dfrac{\sqrt{1+ \sin x} + \sqrt{1- \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right) = Ax, where 0<x<π20 < x < \dfrac{\pi}{2}, find the value of AA.

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