If `x+y+z=a`, show that`(1/x+1/y+1/z)^-1ge9/a`

If xyz=1 then show that (1+x+y^(-1))^(-1)+(1+y+z^(-1))^(-1)+(1+z+x^(-1))^(-1)=1

If xy + yz + zx = 1 , show that x/(1-x^(2)) +y/(1-y^(2)) + z/(1-z^(2))= (4xyz)/((1-x^(2))(1-y^(2)) (1-z^(2)))

If xy + yz + zx = 1 , show 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

If x+y+z=1a n dx ,y ,z are positive, then show that (x+1/x)^2+(y+1/y)^2+(z+1/z)^2>(100)/3

If xy+yz+zx=1 show 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 = 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)=12^(z) show that (1)/(z)=(1)/(y)+(2)/(x)