$\begin{array}{l} A = \dfrac{{\sqrt {\sqrt 5 + 2} + \sqrt {\sqrt 5 - 2} }}{{\sqrt {2\sqrt 5 + 2} }}\\ {A^2} = \dfrac{{\sqrt 5 + 2 + \sqrt 5 - 2 + 2\sqrt {\left( {\sqrt 5 + 2} \right)\left( {\sqrt 5 - 2} \right)} }}{{2\sqrt 5 + 2}}\\ {A^2} = \dfrac{{2\sqrt 5 + 2\sqrt {5 - 4} }}{{2\sqrt 5 + 2}} = \dfrac{{2\sqrt 5 + 2}}{{2\sqrt 5 + 2}} = 1\\ \Rightarrow A = 1 \end{array}$
