$A = \left(\sqrt{6+\sqrt{20}}-2\sqrt{3-\sqrt{5}}+\sqrt{15-10\sqrt{2}}\right)\ :\ \left(2+\sqrt8\right)$
$\to A = \left(\sqrt{\left(\sqrt5\right)^2 +2.\sqrt{5}.1 + 1^2}-\sqrt2.\sqrt{\left(\sqrt5\right)^2-2.\sqrt{5}.1+ 1^2}+\sqrt{\left(\sqrt{10}\right)^2-2.\sqrt{10}\sqrt{5} + \left(\sqrt5\right)^2}\right)\ :\ \left[2\left(1+\sqrt2\right)\right]$
$\to A = \left[\sqrt{\left(\sqrt5+1\right)^2}-\sqrt2.\sqrt{\left(\sqrt5-1\right)^2}+\sqrt{\left(\sqrt{10}-\sqrt5\right)^2}\right]\ :\ \left[2\left(1+\sqrt2\right)\right]$
$\to A = \left[\sqrt5 +1 - \sqrt2\left(\sqrt5-1\right) + \left(\sqrt{10}-\sqrt5\right)\right]\ :\ \left[2\left(1+\sqrt2\right)\right]$
$\to A = \left(1+\sqrt2\right)\ :\ \left[2\left(1+\sqrt2\right)\right]$
$\to A =\dfrac12$
