Complicated Complex Fractions

Let $S$ be the desired value. Note that

$xy + z - 1 = xy + 1 - x - y = (x - 1)(y - 1)$

from substitution. Likewise,

$yz +z - 1 = (y - 1)(z - 1)$

and

$zx + y - 1 = (z - 1)(x - 1)$ .

Hence,

$S = |\frac {1}{(x - 1)(y - 1)} + \frac {1}{(y - 1)(z - 1)} + \frac {1}{(z - 1)(x - 1)}|$

$S = |\frac {x + y + z - 3}{(x - 1)(y - 1)(z - 1)} = \frac {-1}{(x - 1)(y - 1)(z - 1)}|$

$S = |\frac {-1}{xyz - (xy + yz + zx) + x + y + z - 1}|$

$S = |\frac {-1}{5 - (xy + yz + zx)}|$

But, $2(xy + yz + zx) = (x + y + z)^2 - (x^2 + y^2 + z^2) = 1$ . This means $xy + yz + zx = \frac {1}{2}$ . I we input this value into the equation, we get an answer of

$S = |\frac {-1}{5 - \frac {1}{2}}| = |\frac {-2}{9}| = 0.222...$