<s>[INST] 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
$$
\frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1
$$
 [/INST] 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?
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