Simplifying an expression with a nested radical

Question

I'm doing some work on functions and have come across a problem in which I need to simplify a certain expression. I can't move any further on because I'm unsure of how to simplify this expression!

Here it is: $$ \frac{1}{\sqrt{1+\sqrt{x^2-1}}} $$

Any help is appreciated, thanks so much!

Here is a tutorial on how to typeset mathematics on this site. — N. F. Taussig, Apr 04 '17 at 14:27
On that post, I wrote a comment exactly like N.F Taussig. Deja vu! — jonsno, Apr 04 '17 at 14:32

score 0 · Accepted Answer · answered Apr 04 '17 at 14:40

I don't think there is much that can be done. If you want to avoid the "square root inside square root", you can expand the fraction with $\sqrt{1-\sqrt{x^2-1}}$ or $\sqrt{\sqrt{x^2-1} - 1}$, depending on which version produces a positive discriminant:

$$ \frac{1}{\sqrt{1+\sqrt{x^2-1}}} = \frac{\sqrt{1-\sqrt{x^2-1}}}{\sqrt{2-x^2}} $$

Simplifying an expression with a nested radical

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