Simplify: $\;$ $\left[\dfrac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}} - \dfrac{m - \sqrt{m^2 - n^2}}{m + \sqrt{m^2 - n^2}}\right] \times \dfrac{n^2}{4m \sqrt{m^2 - n^2}}$
$\left[\dfrac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}} - \dfrac{m - \sqrt{m^2 - n^2}}{m + \sqrt{m^2 - n^2}}\right] \times \dfrac{n^2}{4m \sqrt{m^2 - n^2}}$
$= \left[\dfrac{\left(m + \sqrt{m^2 - n^2}\right)^2 - \left(m - \sqrt{m^2 - n^2}\right)^2}{\left(m + \sqrt{m^2 - n^2}\right) \left(m - \sqrt{m^2 - n^2}\right)}\right] \times \dfrac{n^2}{4m \sqrt{m^2 - n^2}}$
$= \left[\dfrac{\left(m + \sqrt{m^2 - n^2} + m - \sqrt{m^2 - n^2}\right) \left(m + \sqrt{m^2 - n^2} - m + \sqrt{m^2 - n^2}\right)}{m^2 - \left(m^2 - n^2\right)}\right]$
$\hspace{10cm} \times \dfrac{n^2}{4m \sqrt{m^2 - n^2}}$
$= \left[\dfrac{2m \times 2 \sqrt{m^2 - n^2}}{n^2}\right] \times \dfrac{n^2}{4m \sqrt{m^2 - n^2}}$
$= 1$