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