<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] $$x(1 - y^2)(1 - z^2) + y(1 - x^2)(1 - z^2) + z(1 - x^2)(1 - y^2)$$
$$=$$
$$x - xy^2 - xz^2 + xy^2z^2 + y - yx^2 - yz^2 + yx^2z^2 + z - zx^2 - zy^2 + zx^2y^2$$
$$=$$
$$(x + y + z)\cdot 1 - xy^2 - xz^2  - yx^2 - yz^2 - zx^2 - zy^2 + \frac{xy^2z^2 + yx^2z^2 + zx^2y^2}{1}$$
$$=$$
$$4xyz$$
</s>