Prove that : `tan(1/2){sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))}=(x+y)/(1-x y)` , if `|x|<1,` `y >0` and `x y<1` .

Explanation

The given function is `\tan \frac{1}{2}[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}]`




`\therefore \tan \frac{1}{2}[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}] `




`\tan \frac{1}{2}[2 \tan ^{-1} x+2 \tan ^{-1} y] [as 2 \tan ^{-1} x=\sin ^{-1}` `\frac{2 x}{1+x^{2}}=\cos ^{-1} \frac{1-x^{2}}{1+x^{2}}]`



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