$\begin{array}{l} \sqrt {3 + \sqrt {5 + 2\sqrt 3 } } .\sqrt {3 - \sqrt {5 + 2\sqrt 3 } } \\ = \sqrt {\left( {3 + \sqrt {5 + 2\sqrt 3 } } \right)\left( {3 - \sqrt {5 + 2\sqrt 3 } } \right)} \\ = \sqrt {9 - \left( {5 + 2\sqrt 3 } \right)} = \sqrt {4 - 2\sqrt 3 } = \sqrt {{{\left( {\sqrt 3 - 1} \right)}^2}} \\ = \sqrt 3 - 1 \end{array}$
