[tex]\frac{\sqrt{1 - x} + \frac{1}{\sqrt{1 + x}}}{1 + \frac{1}{\sqrt{1 - x^2}}} = \frac{\sqrt{1 - x} + \frac{1}{\sqrt{1 + x}}}{1 + \frac{1}{\sqrt{1 - x^2}}} \cdot \frac{1 - \frac{1}{\sqrt{1 - x^2}}}{1 - \frac{1}{\sqrt{1 - x^2}}} = \frac{\sqrt{1 - x} - \frac{\sqrt{1-x}}{\sqrt{1-x^2}} + \frac{1}{\sqrt{1+x}} - \frac{1}{\sqrt{(1+x)(1-x^2)}}}{1 - \frac{1}{1 - x^2}}.[/tex]

Note que

[tex]\sqrt{1 - x} - \frac{\sqrt{1-x}}{\sqrt{1-x^2}} + \frac{1}{\sqrt{1+x}} - \frac{1}{\sqrt{(1+x)(1-x^2)}} = \sqrt{1 - x} + \frac{-\sqrt{1-x}\sqrt{1+x} + \sqrt{1-x^2}}{\sqrt{1-x^2}\sqrt{1+x}} - \frac{1}{\sqrt{(1+x)(1-x^2)}} = \sqrt{1 - x} + \frac{-\sqrt{1-x^2} + \sqrt{1-x^2}}{\sqrt{1-x^2}\sqrt{1+x}} - \frac{1}{\sqrt{(1+x)(1-x^2)}} = \sqrt{1 - x} - \frac{1}{\sqrt{(1+x)(1-x^2)}} = \frac{\sqrt{1-x}\sqrt{(1+x)(1-x^2)} - 1}{\sqrt{(1+x)(1-x^2)}} = \frac{-x^2}{\sqrt{(1+x)(1-x^2)}}.[/tex]

[tex]1-\frac{1}{1-x^2} = \frac{1-x^2-1}{1-x^2} = \frac{-x^2}{1-x^2}.[/tex]

Desse modo, obtemos o seguinte

[tex]\frac{\sqrt{1 - x} + \frac{1}{\sqrt{1 + x}}}{1 + \frac{1}{\sqrt{1 - x^2}}} = \frac{\frac{-x^2}{\sqrt{(1+x)(1-x^2)}}}{\frac{-x^2}{1-x^2}} = \frac{1-x^2}{\sqrt{(1+x)(1-x^2)}} = \sqrt{\frac{(1-x^2)^2}{(1+x)(1-x^2)}} = \sqrt{\frac{1-x^2}{1+x}} = \sqrt{\frac{(1-x)(1+x)}{1+x}} = \sqrt{1-x}.[/tex]