What is the simplified value of $$\frac{{\left( {x + y + z} \right)\left( {xy + yz + zx} \right) - xyz}}{{\left( {x + y} \right)\left( {y + z} \right)\left( {z + x} \right)}}?$$

$$\eqalign{ & \frac{{\left( {x + y + z} \right)\left( {xy + yz + zx} \right) - xyz}}{{\left( {x + y} \right)\left( {y + z} \right)\left( {z + x} \right)}} \cr & {\text{put }}z = 0, \cr & = \frac{{\left( {x + y + 0} \right)\left( {xy + 0 + 0} \right) - 0}}{{\left( {x + y} \right)\left( {y + 0} \right)\left( {0 + x} \right)}} \cr & = \frac{{\left( {x + y} \right)\left( {xy} \right)}}{{\left( {x + y} \right)\left( {xy} \right)}} \cr & = 1 \cr} $$