Find the value of: `tan(1/2 [sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))]),|x|<1, y >0`and `x y < 1`

Explanation

To solve the problem, we need to find the value of:

\[ \tan\left(\frac{1}{2} \left[\sin^{-1}\left(\frac{2x}{1+x^2}\right) + \cos^{-1}\left(\frac{1-y^2}{1+y^2}\right)\right]\right) \]

Given the conditions \( |x| < 1 \), \( y > 0 \), and \( xy < 1 \).

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