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:
- Substitution of $x = \sqrt{5}$
- Substitution of $x = 2\sqrt{5}$
- Substitution of $x = 5+\sqrt{5}$
- Substitution of $x = \sqrt{5 + \sqrt{5}}$
- Manipulations by substituting the golden ratio
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.
Hint $\ $ Let $\ a = \sqrt{5+2\sqrt 5},\ b = \sqrt{10+2\sqrt 5}.\,$ Show $\,\color{#c00}{b = (\sqrt5 -1) a},\,$ so scaling the top and bottom of the fraction by $\,8\,$ yields $\ \dfrac{4a+ 2\sqrt 5 a - \color{#c00}{(\sqrt5 -1) a}}{10a + 2\sqrt 5 a}^{\phantom{1^{1^1}}}\!\!\!\!\!\! =\, \dfrac{1}2$