<s>[INST] If $xy+yz+zx=1$ , show that $\dfrac x{1-x^2}+\dfrac y{1-y^2}+\dfrac z{1-z^2}=\dfrac{4xyz}{(1-x^2)(1-y^2)(1-z^2)}$. It is to be solved using trigonometry. I tried taking $x$, $y$ and $z$ as $\sin(a)$, $\sin(b)$ and $\sin(c)$ respectively, but could get no further.
 [/INST] Let $x=\tan A, y = \tan B, z = \tan C$. Then 
\begin{align*}
\tan(A+B+C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1-\tan A \tan B -\tan B \tan  C-\tan C \tan A} 
\end{align*}
Hence $A+B+C $ is an odd multiple of $\frac{\pi}{2}$. We need to prove
\begin{align*}
\tan 2A + \tan 2B + \tan 2C  = \tan 2A \tan 2B \tan 2C
\end{align*}
This is clear since $2A + 2B + 2C$ is a multiple of $\pi$
</s>