Chứng minh rằng:

\(\frac{{x - y}}{{1 + xy}} + \frac{{y - z}}{{1 + yz}} + \frac{{z - x}}{{1 + zx}} = \frac{{x - y}}{{1 + xy}} \cdot \frac{{y - z}}{{1 + yz}} \cdot \frac{{z - x}}{{1 + zx}}\).

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

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

\( = \left( {x - y} \right)\left( {\frac{1}{{1 + xy}} - \frac{1}{{1 + yz}}} \right) + \left( {x - z} \right)\left( {\frac{1}{{1 + yz}} - \frac{1}{{1 + zx}}} \right)\)

\[ = \left( {x - y} \right)\frac{{yz - xy}}{{\left( {1 + xy} \right)\left( {1 + yz} \right)}} + \left( {x - z} \right)\frac{{zx - yz}}{{\left( {1 + yz} \right)\left( {1 + zx} \right)}}\]

\[ = \frac{{y\left( {x - y} \right)\left( {z - x} \right)}}{{\left( {1 + xy} \right)\left( {1 + yz} \right)}} + \frac{{z\left( {x - y} \right)\left( {x - z} \right)}}{{\left( {1 + yz} \right)\left( {1 + zx} \right)}}\]

\[ = \frac{{y\left( {x - y} \right)\left( {z - x} \right)}}{{\left( {1 + xy} \right)\left( {1 + yz} \right)}} - \frac{{z\left( {x - y} \right)\left( {z - x} \right)}}{{\left( {1 + yz} \right)\left( {1 + zx} \right)}}\]

\[ = \frac{{\left( {x - y} \right)\left( {z - x} \right)}}{{1 + yz}} \cdot \frac{{y\left( {1 + zx} \right) - z\left( {1 + xy} \right)}}{{\left( {1 + xy} \right)\left( {1 + zx} \right)}}\]

\[ = \frac{{\left( {x - y} \right)\left( {z - x} \right)}}{{1 + yz}} \cdot \frac{{y - z}}{{\left( {1 + xy} \right)\left( {1 + zx} \right)}}\]

\( = \frac{{x - y}}{{1 + xy}} \cdot \frac{{y - z}}{{1 + yz}} \cdot \frac{{z - x}}{{1 + zx}}\).