Alg3.3.9

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Show that \frac{1}{1+x^{a-b}+x^{a-c}}+\frac{1}{1+x^{b-c}+x^{b-a}}+\frac{1}{1+x^{c-a}+x^{c-b}}=1\,

Simplifying each fraction of the left hand side,

\frac{1}{1+x^{a-b}+x^{a-c}}=\frac{1}{1+\frac{x^a}{x^b}+\frac{x^a}{x^c}}=\frac{x^{b+c}}{x^{b+c}+x^{a+c}+x^{a+b}}\,

Similarly the next two fractions are simplified as below

\frac{1}{1+x^{b-c}+x^{b-a}}=\frac{x^{c+a}}{x^{c+a}+x^{a+b}+x^{b+c}}\,

\frac{1}{1+x^{c-a}+x^{c-b}}=\frac{x^{a+b}}{x^{a+b}+x^{c+b}+x^{c+a}}\,

Adding the three,we get

\frac{1}{1+x^{a-b}+x^{a-c}}+\frac{1}{1+x^{b-c}+x^{b-a}}+\frac{1}{1+x^{c-a}+x^{c-b}}=\frac{x^{b+c}}{x^{b+c}+x^{a+c}+x^{a+b}}+\frac{x^{c+a}}{x^{c+a}+x^{a+b}+x^{b+c}}+\frac{x^{a+b}}{x^{a+b}+x^{c+b}+x^{c+a}}\,

Simplifying the right hand side expression,we get

\frac{x^{a+b}+x^{b+c}+x^{c+a}}{x^{a+b}+x^{b+c}+x^{c+a}}=1\,


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