<s>[INST] tough algebric problem? I wanted to know how can i prove that if

$xy+yz+zx=1$, then
$$ \frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2}
     = \frac{2}{\sqrt{(1+x^2)(1+y^2)(1+z^2)}}$$
I did let $x=\tan A$,  $y=\tan B$,   $z=\tan C$

given $xy+yz+zx =1$ we have $\tan A \tan B+ \tan B \tan C+\tan C \tan A=1$
$\tan C(\tan A+\tan B)=1-\tan A \tan B$, or $\tan(A+B)=\tan(\pi/2 -C)$ we have $A+B+C=\pi/2$.
what to do now?  
Any help appreciated.
thanks.
 [/INST] HINT:
$$\frac x{1+x^2}=\frac {\tan A}{1+\tan^2A}=\frac{2\sin A\cos A}2=\frac{\sin2A}2$$
Now , $$\sin2A+\sin2B+\sin2C=2\sin(A+B)\cos(A-B)+2\sin C\cos C$$
$$=2\sin\left(\frac\pi2-C\right)\cos(A-B)+2\sin C\cos C$$
$$=2\cos C\{\cos(A-B)+\cos(A+B)\}$$ as $\sin C=\sin\{\frac\pi2-(A+B)\}=\cos(A+B)$
$$\implies \sin2A+\sin2B+\sin2C=2\cos C\cdot2\cos A\cos B$$
and  $$\frac1{\sqrt{1+x^2}}=\frac1{\sec A}=\cos A$$
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