Mathematic Problems
$\left(\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}+\frac{1}{1+\frac{b}{a}}-\frac{1}{1-\frac{b}{a}}\right):\frac{1-\frac{a-3b}{a+b}}{\frac{3a+b}{a-b}-3}$
$\left(\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}{+}\frac{1}{1+\frac{b}{a}}{-}\frac{1}{1-\frac{b}{a}}\right):\frac{1-\frac{a-3b}{a+b}}{\frac{3a+b}{a-b}-3}={A}:{B}$
${A}{=}\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}{+}\frac{1}{1+\frac{b}{a}}{-}\frac{1}{1-\frac{b}{a}}$
${A}=\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}+\frac{1}{1+\frac{b}{a}}-\frac{1}{1-\frac{b}{a}}$
$=\frac{\frac{{a}^{2}+{b}^{2}}{{a}{b}}}{\frac{{a}^{2}-{b}^{2}}{{a}{b}}}+\frac{\frac{1}{1}}{\frac{a+b}{a}}-\frac{\frac{1}{1}}{\frac{a-b}{a}}$
$=\frac{{a}^{2}+{b}^{2}}{{a}^{2}-{b}^{2}}+\frac{a}{a+b}-\frac{a}{a-b}$
$=\frac{{a}^{2}+{b}^{2}+a\left(a-b\right)-a\left(a+b\right)}{\left(a+b\right)·\left(a-b\right)}$
$=\frac{{a}^{2}+{b}^{2}{+}{{a}}^{2}-ab{-}{{a}}^{2}-ab}{\left(a+b\right)·\left(a-b\right)}$
$=\frac{{a}^{2}+{b}^{2}-2ab}{\left(a+b\right)·\left(a-b\right)}$
$=\frac{{\left(a-b\right)}^{{2}}}{\left(a+b\right)·{\left(}{a}{-}{b}{\right)}}$
$=\frac{a-b}{a+b}$
${B}{=}\frac{\frac{a+b-a+3b}{a+b}}{\frac{3a+b-3a+3b}{a-b}}$
${B}=\frac{\frac{a+b-a+3b}{a+b}}{\frac{3a+b-3a+3b}{a-b}}$
$=\frac{\frac{{4}{b}}{a+b}}{\frac{{4}{b}}{a-b}}$
$=\frac{a-b}{a+b}$
${A}:{B}=\frac{a-b}{a+b}:\frac{a-b}{a+b}$
${A}:{B}=\frac{a-b}{a+b}:\frac{a-b}{a+b}$
$=\frac{{a}{-}{b}}{{a}{+}{b}}·\frac{{a}{+}{b}}{{a}{-}{b}}$
$=1$
$\left(\frac{\frac{a}{b}+\frac{b}{a}}{\frac{a}{b}-\frac{b}{a}}+\frac{1}{1+\frac{b}{a}}-\frac{1}{1-\frac{b}{a}}\right):\frac{1-\frac{a-3b}{a+b}}{\frac{3a+b}{a-b}-3}=1$