$\begin{align*}
\sqrt{x+\sqrt{x^2-1}}+\sqrt{x-\sqrt{x^2-1}} &= \sqrt{\left( \sqrt{x+\sqrt{x^2-1}}+\sqrt{x-\sqrt{x^2-1}} \right)^2} \\
&= \sqrt{x+\sqrt{x^2-1}+x-\sqrt{x^2-1}+\2\sqrt{(x+\sqrt{x^2-1} = (x-\sqrt{x^2-1})}} \\
&= \sqrt{2x+2\sqrt{x^2-(x^2-1)}} = \sqrt{2x+2} = \sqrt{2} \sqrt{x+1}
\end{align*}$