If `x+y+z=0`,then prove that: `(1)/(1+(1)/(a^(z))+a^(x))+(1)/(1+(1)/(a^(x))+a^(y))+(1)/(1+(1)/(a^(y))+a^(z))=1`

Show that: ((a^(x+1))/(a^(y+1)))^(x+y)((a^(y+2))/(a^(z+2)))^(y+z)((a^(z+3))/(a^(x+3)))^(z+x)=1

If x,y,z are all different real numbers,then prove that (1)/((x-y)^(2))+(1)/((y-z)^(2))+(1)/((z-x)^(2))=((1)/(x-z)+(1)/(y-z)+(1)/(z-x))^(2)

If xy+yz+xz=1, then prove that (x)/(1-x^(2))+(y)/(1-y^(2))+(z)/(1-z^(2))=(4xyz)/((1-x^(2))(1-y^(2))(1-z^(2)))

If x,y,z gt 0 then prove that (x+y+z)(1/x+1/y+1/z) geq 9

Given x+ (1)/(y)=1 and y + (1)/(z)=1 find (x+y+z) + ((1)/(x) + (1)/(y) + (1)/(z))= ?

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 .

If 2^(x)=3^(y)=6^(-z) prove that (1)/(x)+(1)/(y)+(1)/(z)=0

(1)/((x-y)^(2))+(1)/((y-z)^(2))+(1)/((z-x)^(2))=((1)/(x-y)+(1)/(y-z)+(1)/(z-x))^(2)

If x+y+z=xzy , prove that : (2x)/(1-x^2) + (2y)/(1-y^2) + (2z)/(1-z^2) = (2x)/(1-x^2) (2y)/(1-y^2) (2z)/(1-z^2)

If x,y,z gt 0 and x + y + z = 1, the prove that (2x)/(1 - x) + (2y)/(1 - y) + (2z)/(1 - z) ge 3 .