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

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

If x!=y!=z and |x x^2 1+x^3 y y^2 1+y^3 z z^2 1+z^3|=0 , then prove that x y z=-1 .

If x,y,z are different and Delta=det[[x,x^(2),1+x^(3)y,y^(2),1+y^(3)z,z^(2),1+z^(3)]]=0, then prove that 1+xyz=0,

If x+y+z=x y z 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)dot

If x+y+z=x y z 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)dot

If x > 0, y >0, z > 0 and x +y + z = 1 , prove that (1+x)(1+y)(1+z)ge 8(1-x) (1-y) (1-z)