If `x y+y z+x z=1`,then prove that `x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=(4x y z)/((1-x^2)(1-y^2)(1-z^2)`

Explanation

As x,y,z are real, we can find numbers tanA, tanB, tanC such that
x=tanA
y=tanB
z=tanC
The given condition `xy+yz+zx=1` now reads
`tanAtanB+tanBtanC+tanAtanC=1`
`rArr tanAtanB+tanC(tanA+tanB)=1`
`rArr tanC=(1-tanAtanB)/(tanA+tanB)`
`rArr cotC=(tanA+tanB)/(1-tanAtanB)=tan(A+B)`
`rArr tan(pi/2-C)=tan(A+B)`
Now, `x/(1-x^(2)) + y/(1-y^(2)) + z/(1-z^(2))= sum (tanA)/(1-tan^(2)A) = 1/2sum (2tanA)/(1-tan^(2)A)=1/2 sum tan2A`.
Now, A+B+C`=pi/2`
`rArr 2A+ 2B +2C=pi`
For example, we have
`tan2A+tan2B+tan2C=tan2A+tan2B.tan2C`
`rArr (2x)/(1-x^(2)) + (2y)/(1-y^(2)) + (2z)/(1-y^(2)) = ((2y)/(1-x^(2)) (2y)/(1-y^(2)) (2z)/(1-z^(2)))`
`rArr x/(1-x)^(2) + y/(1-y^(2)) + z/(1-z^(2)) = (4xyz)/((1-x^(2))(1-y^(2))(1-z^(2))`