# Show $\frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1$ given $xyz = 1$

Please help me prove the equality: If $xyz=1$, prove that

$$\frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1$$

Note that $$\dfrac{x}{xy+x+1} = \dfrac{xz}{xyz+xz + z} = \dfrac{xz}{xz+z+1}$$ Similarly, we get $$\dfrac{y}{yz+y+1} = \dfrac1{z+1+\dfrac1y} = \dfrac1{z+1+xz}$$ Now add the three terms and finish it off by canceling the numerator and denominator.
Hint: Rewrite the $1$ in the first denominator as $xyz,$ so that the left hand side becomes \begin{align}\frac1{y+1+yz}+\frac{y}{yz+y+1}+\frac{z}{xz+z+1} &= \frac{1+y}{yz+y+1}+\frac{z}{xz+z+1}\\ &= 1-\frac{yz}{yz+y+1}+\frac{z}{xz+z+1}.\end{align} Can you go from here?