$\text{ If }xy+yz+zx=1\text{ then }\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}=$

$xy+yz+zx=1$
$\text{ Put }x=\tan A,y=\tan B,Z=\tan C$
$\tan A \tan B+\tan B\tan C+\tan c\tan A=1$
$$\begin{gathered} A+B+C=\frac{\mathrm{\pi }}{2} \\ \Rightarrow 2A+2B+2C=\mathrm{\pi } \end{gathered}$$
$$\begin{gathered} \sin 2A+\sin 2B+\sin 2C=4\cos AcosBcosC \\ \Rightarrow \frac{2\tan A}{1+{\tan }^{2}A}+\frac{2\tan B}{1+{\tan }^{2}B}+\frac{2\tan C}{1+{\tan }^{2}C}=\frac{4}{secA secB secC} \\ \Rightarrow \frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}=\frac{2}{\sqrt{\left(1+x^2\right)\left(1+y^2\right)\left(1+z^2\right)}} \end{gathered}$$