In the above graph, the red curve is a part of the curve \(x^2-y^2=1\) (mostly in the first quadrant). A point \(A\) lies on this red curve and is connected to the origin by the green line segment.

If the \(x\)-coordinate of point \(A\) equals \(\dfrac{e^{2016}+e^{-2016}}{2}\), find the area bounded by the red curve, green line segment and the black \(x\)-axis in the first quadrant.

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