Find the value of $\tan\large\frac{1}{2}$$\bigg[ \sin^{-1}\large \frac{2x}{1+x^2}$$+\cos^{-1}\large\frac{1-y^2}{1+y^2} \bigg].$$\: where \: |x| < 1, y > 0 \: and \: xy < 1.$

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