\frac{1}{2} \cdot \frac{\sqrt{x^2-1} }{ \sqrt{x+\sqrt{x^2-1}}+\sqrt{x-\sqrt{x^2-1}}} =
\frac{1}{2} \cdot \frac{\sqrt{(x+1)(x-1)} }{\sqrt{2} \sqrt{x+1}} =
\frac{\sqrt{2}}{4} \cdot\sqrt{x-1}