Simplify: $\;$ $\dfrac{\dfrac{1}{\sqrt{a - 1}} + \sqrt{a + 1}}{\dfrac{1}{\sqrt{a + 1}} - \dfrac{1}{\sqrt{a - 1}}} : \dfrac{\sqrt{a + 1}}{\left(a - 1\right) \sqrt{a + 1} - \left(a + 1\right) \sqrt{a - 1}}$
$\dfrac{\dfrac{1}{\sqrt{a - 1}} + \sqrt{a + 1}}{\dfrac{1}{\sqrt{a + 1}} - \dfrac{1}{\sqrt{a - 1}}} : \dfrac{\sqrt{a + 1}}{\left(a - 1\right) \sqrt{a + 1} - \left(a + 1\right) \sqrt{a - 1}}$
$= \left\{\dfrac{\dfrac{1 + \sqrt{a + 1} \sqrt{a - 1}}{\sqrt{a - 1}}}{\dfrac{\sqrt{a - 1} - \sqrt{a + 1}}{\sqrt{a + 1} \sqrt{a - 1}} } \right\} \times \left\{\dfrac{\left(a - 1\right) \sqrt{a + 1} - \left(a + 1\right) \sqrt{a - 1}}{\sqrt{a + 1}} \right\}$
$= \left\{\dfrac{\left[1 + \sqrt{a + 1} \sqrt{a - 1}\right] \sqrt{a + 1}}{\sqrt{a - 1} - \sqrt{a + 1}} \right\} \times \left\{\dfrac{\left(a - 1\right) \sqrt{a + 1} - \left(a + 1\right) \sqrt{a - 1}}{\sqrt{a + 1}} \right\}$
$= \left\{\dfrac{1 + \sqrt{a + 1} \sqrt{a - 1}}{\sqrt{a - 1} - \sqrt{a + 1}} \right\} \times \left\{\sqrt{a - 1} \sqrt{a + 1} \left(\sqrt{a - 1} - \sqrt{a + 1}\right) \right\}$
$= \left\{1 + \sqrt{a + 1} \sqrt{a - 1} \right\} \times \sqrt{a - 1} \sqrt{a + 1}$
$= \sqrt{a - 1} \sqrt{a + 1} + \left(a + 1\right) \left(a - 1\right)$
$= \sqrt{a^2 - 1} + a^2 - 1$