`tan^(- 1)x+tan^(- 1)y=1/2sin^(- 1)(2(x+y)(1-x y))/((1+x^2)(1+y^2))`

Prove that: tan^(-1)((1-x)/(1+x))-tan^(-1)((1-y)/(1+y))=sin^(-1)((y-x)/(sqrt((1+x^(2))(1+y^(2)))))

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| 0 and x y<1 .

Prove that tan{1/2sin^(-1)((2x)/(1+x^(2)))+1/2cos^(-1)((1-y^(2))/(1+y^(2)))}=(x+y)/(1-xy) , where |x| lt 1,y gt 0 and xy lt 1 .

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

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

Simplity tan{1/2sin^(-1)((2x)/(1+x^(2)))+1/2cos^(-1)(1-y^(2))/(1+y^(2))} , if xgtygt1 .

If tan^(-1)x,tan^(-1)y,tan^(-1)z are in A.P the (2y)/(1-y^(2))=

(1/y^2 \ ((cos(tan^(-1) y) + y sin(tan^(-1) y))/(cot(sin^(-1) y) + tan(sin^(-1) y)) )^2 + y^4)^(1//2) takes value