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

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

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

Prove the following "tan"^(-1)((1-x)/(1+x))-"tan"^(-1)((1-y)/(1+y))="sin"^(-1)((y-x)/(sqrt(1+x^(2))sqrt(1+y^(2)))) .

If the points (x_(1),y_(1)),(x_(2),y_(2)), and (x_(3),y_(3)) are collinear show that (y_(2)-y_(3))/(x_(2)x_(3))+(y_(3)-y_(1))/(x_(3)x_(1))+(y_(1)-y_(2))/(x_(1)x_(2))=0

If x + y + z = xyz , prove that x(1 -y^(2)) (1- z^(2))+ y(1- z^(2))(1- x^(2)) +z(1-x^(2)) (1- y^(2)) = 4xyz .

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