<s>[INST] If $x+y+z=xyz$, find $\frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}$ I found this question in a maths worksheet of trigonometry (kinda odd, right?), but I dont know how to figure it out.
If $\displaystyle x+y+z=xyz$, find $\displaystyle\frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}$
First I thought of taking x,y and z as $\displaystyle \tan A, \tan B,$ and $\tan C,$ making $A+B+C=\pi$, but couldnt solve ahead. Any solution not involving trigonometry would do as well.
I also think that this question does not even relate to trigo.....or does it?
 [/INST] Case I: At least one of $x^2$, $y^2$ and $z^2$ is equal to $1/3$. This case is possible, e.g., $x=\frac{\sqrt{3}}{3}=-y$ and $z=0$. In this case, the formula $\displaystyle \frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}$ does not make any sense.
Case II: None of $x^2$, $y^2$ and $z^2$ is equal to $1/3$ and $xyz=0$. For example, we say $x=-y$ and $z=0$, which implies that
\begin{eqnarray*}
\frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}=\frac{3x-x^3}{1-3x^2}+\frac{-3x+x^3}{1-3x^2}+0=0.
\end{eqnarray*}
Case III: None of $x^2$, $y^2$ and $z^2$ is equal to $1/3$ and $xyz\neq0$. Take $x=\tan A$, $y=\tan B$ and $z=\tan C$ for some $A,B,C\in(-\pi/2,\pi/2)$, since $x+y+z=xyz$, then $xy,xz,yz\neq 1$, which implies that $A+B$, $A+C$ and $B+C$ can not be $\pm\pi/2$.
Claim I: $A+B+C\neq\pm\pi/2$.
If not, then $\tan(A+B)=\cot C$, that is, $\displaystyle\frac{x+y}{1-xy}=\frac{1}{z}$, that is, $(x+y)z=1-xy$. Hence $(x+y)z^2=z-xyz=z-(x+y+z)=-(x+y)$, that is, $x+y=0$. Since $x+y+z=xyz$ and $z\neq0$, we get $xy=1$, contradiction.
In summary, we can define $\tan(3A)$, $\tan(3B)$, $\tan(3C)$ and $\tan(A+B+C)$. Since $\displaystyle \tan(3\theta)=\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}$, then we have 
\begin{eqnarray*}
\frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}=\tan(3A)+\tan(3B)+\tan(3C).
\end{eqnarray*}
Notice that
\begin{eqnarray*}
\tan(A+B+C)&=&\frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan A\tan C-\tan B\tan C}\\
&=&\frac{x+y+z-xyz}{1-xy-xz-yz}=0.
\end{eqnarray*}
Then there exists some $m\in\{-1,0,1\}$ such that $A+B+C=m\pi$. Hence $\tan(A+B)+\tan C=0$, which implies that $\tan^2(A+B)\neq1/3$. Hence we can define $\tan(3A+3B)$. So we have
\begin{eqnarray*}
0&=&\frac{\tan(3A)+\tan(3B)+\tan(3C)-\tan(3A)\tan(3B)\tan(3C)}{1-\tan(3A)\tan(3B)-\tan(3A)\tan(3C)-\tan(3B)\tan(3C)}.
\end{eqnarray*}
Then we get
\begin{eqnarray*}
\frac{3x-x^3}{1-3x^2}+\frac{3y-y^3}{1-3y^2}+\frac{3z-z^3}{1-3z^2}=\tan(3A)+\tan(3B)+\tan(3C)=\tan(3A)\tan(3B)\tan(3C).
\end{eqnarray*}
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