Prove that `((x^(-1)+y^(-1)))/(x^(-1))+((x^(-1)+y^(-1)))/(y^(-1))=(x+y)^2/(xy)`

Prove that ((x^(-1)+y^(-1))/x^(-1))^(-1)+((x^(-1)-y^(-1))/x^(-1))^(-1)=(2y^(2))/(y^(2)-x^(2))

underset Prove that ((x ^ (- 1) + y ^ (- 1)) / (x ^ (- 1))) ^ (- 1) + ((x ^ (- 1) -y ^ (- 1) ) / (x ^ (- 1))) ^ (- 1) = (2y ^ (2)) / (y ^ (2) -x ^ (2))

Prove that : tan^(-1) x+cot^(-1) y = tan^(-1) ((xy+1)/(y-x))

Prove that : tan^(-1) x+cot^(-1) y = tan^(-1) ((xy+1)/(y-x))

Prove that tan^(-1)(1/(x+y))+tan^(-1)(y/(x^2+xy+1) )= cot^(-1)x .

Prove that tan^(-1)x-tan^(-1)y=tan^(-1)((x-y)/(1+xy)),xygt-1

Prove that tan ^(-1)""(1)/(x+y)+ tan ^(-1)""(y)/(x^(2)+xy+1)= cot ^(-1)x.

What is the expression (x+y)^(-1) (x^(-1) +y^(-1)) (xy^(-1) +x^(-1)y)^(-1) equal to:

If S_(n)=(x+y)+(x^(2)+xy+y^(2))+(x^(3)+x^(2)y+y^(2)x+y^(3))+…n terms then prove that (x-y)S_(n)=[(x^(2)(x^(n)-1))/(x-1)-(y^(2)y^(n)-1)/(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 .