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.
