Simplify: $\;$ $\left[\dfrac{1 + \sqrt{1 - x}}{1 - x + \sqrt{1 - x}} + \dfrac{1 - \sqrt{1 + x}}{1 + x - \sqrt{1 + x}}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$, $\;\;$ $0 < x < 1$
$\left[\dfrac{1 + \sqrt{1 - x}}{1 - x + \sqrt{1 - x}} + \dfrac{1 - \sqrt{1 + x}}{1 + x - \sqrt{1 + x}}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$, $\;\;$ $0 < x < 1$
$= \left[\dfrac{1 + \sqrt{1 - x}}{\sqrt{1 - x} \left(\sqrt{1 - x} + 1\right)} + \dfrac{1 - \sqrt{1 + x}}{\sqrt{1 + x} \left(\sqrt{1 + x} - 1\right)}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$
$= \left[\dfrac{1}{\sqrt{1 - x}} - \dfrac{1}{\sqrt{1 + x}}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$
$= \left[\dfrac{\sqrt{1 + x} - \sqrt{1 - x}}{\left(1 - x\right) \left(\sqrt{1 + x}\right)}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$
$= \left[\dfrac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 - x^2}}\right]^2 \left(\dfrac{x^2 - 1}{2}\right) + 1$
$= \left[\dfrac{1 + x + 1 - x - 2 \sqrt{\left(1 + x\right) \left(1 - x\right)}}{1 - x^2}\right] \left(\dfrac{x^2 - 1}{2}\right) + 1$
$= \left[\dfrac{2 - 2 \sqrt{1 - x^2}}{1 - x^2}\right] \left(\dfrac{x^2 - 1}{2}\right) + 1$
$= \sqrt{1 - x^2} - 1 + 1$
$= \sqrt{1 - x^2}$