\(\dfrac{\dfrac{a-x}{a}+\dfrac{x}{a-x}}{\dfrac{a+x}{a}-\dfrac{x}{a+x}}=\left(\dfrac{a-x}{a}+\dfrac{x}{a-x}\right):\left(\dfrac{a+x}{a}-\dfrac{x}{a+x}\right)\)
\(=\left(\dfrac{\left(a-x\right)\left(a-x\right)}{a\left(a-x\right)}+\dfrac{ax}{a\left(a-x\right)}\right):\left(\dfrac{\left(a+x\right)\left(a+x\right)}{a\left(a+x\right)}-\dfrac{xa}{a\left(a+x\right)}\right)\)
\(=\left(\dfrac{\left(a-x\right)\left(a-x\right)+ax}{a\left(a-x\right)}\right):\left(\dfrac{\left(a+x\right)\left(a+x\right)-ax}{a\left(a+x\right)}\right)\)
\(=\left(\dfrac{a^2-ax+x^2}{a\left(a-x\right)}\right):\left(\dfrac{a^2+ax+x^2}{a\left(a+x\right)}\right)\)
\(=\dfrac{\left(a^2-ax+x^2\right)a\left(a+x\right)}{a\left(a-x\right)\left(a^2+ax+x^2\right)}\)
\(=\dfrac{\left(a^2-ax+x^2\right)\left(a+x\right)}{\left(a^2+ax+x^2\right)\left(a-x\right)}\)
