Radicals and trig

$\small \begin{aligned} \frac {\sqrt{30-6\sqrt{5}}-\sqrt{6+2\sqrt{5}}} {8} & = \left(\frac{\sqrt{3}}{2}\right) \left( \color{#D61F06} {\frac {\sqrt{10-2\sqrt{5}}}{4}} \right) - \left(\frac{1}{2}\right) \left(\color{#3D99F6} {\frac {\sqrt{6+2\sqrt{5}}}{4}} \right) \\ & = \cos{30^\circ}\color{#D61F06}{\sin{36^\circ}} - \sin{30^\circ} \color{#3D99F6} {\cos{36^\circ}} \quad \quad \text{[See Note]} \\ & = \sin{\boxed{6}^\circ} \end{aligned}$

$\color{#D61F06}{\text{Note:}}$

From the identity:

$\small \begin{aligned} \cos{72^\circ} + \cos{144^\circ} & = - \frac{1}{2} \\ \Rightarrow 2 \cos^2{72^\circ} + \cos{72^\circ} -\frac{1}{2} & = 0 \\ 4 \cos^2{72^\circ} + 2\cos{72^\circ} - 1 & = 0 \\ \Rightarrow \cos{72^\circ} & = \frac{\sqrt{5}-1}{4} \\ 2\cos^2{36^\circ} - 1 & = \frac{\sqrt{5}-1}{4} \\ \Rightarrow \color{#3D99F6}{\cos{36^\circ}} & = \sqrt{\frac{\sqrt{5}+3}{8}} = \color{#3D99F6} {\frac{ \sqrt {6+2 \sqrt {5}}}{4}} \\ \Rightarrow \color{#D61F06} {\sin{36^\circ}} & = \sqrt{1 - \left(\frac{ \sqrt {6+2 \sqrt {5}}}{4}\right)^2} = \color{#D61F06} {\frac{\sqrt{10+2 \sqrt{5} } }{4}} \end{aligned}$