MHB How to simplify a diabolical expression involving radicals

[kalish1]

A friend and I have been working on this problem for hours - how can the following expression be simplified analytically?

It equals $\frac{1}{2},$ and we have tried the following to no avail:

1. Substitution of $x = \sqrt{5}$
2. Substitution of $x = 2\sqrt{5}$
3. Substitution of $x = 5+\sqrt{5}$
4. Substitution of $x = \sqrt{5 + \sqrt{5}}$

Here goes:
$$\dfrac{\dfrac{\sqrt{5 + 2\sqrt{5}}}{2} + \dfrac{\sqrt{5(5 + 2\sqrt{5})}}{4} - \dfrac{\sqrt{10 + 2\sqrt{5}}}{8}}{\dfrac{\sqrt{5(5 + 2\sqrt{5})}}{4} + 5 \cdot \dfrac{\sqrt{5 + 2\sqrt{5}}}{4}}$$

Thanks in advance for any help.

This question has been crossposted here - fractions - How to simplify a diabolical expression involving radicals - Mathematics Stack Exchange

 

[Nono713]

Try the substitution $u = \sqrt{5 + 2 \sqrt{5}}$. Then you have:
$$\sqrt{5(5 + 2 \sqrt{5})} = \sqrt{5} u$$
$$\sqrt{10 + 2 \sqrt{5}} = \sqrt{u^2 + 5}$$
You'll still have a root but you'll be able to simplify the fraction I think.