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  1. \[\begin{array}{l}
    \frac{{2 + \sqrt 3 }}{{\sqrt 2 + \sqrt {2 + \sqrt 3 } }} + \frac{{2 – \sqrt 3 }}{{\sqrt 2 – \sqrt {2 – \sqrt 3 } }}\\
    = \frac{{\left( {2 + \sqrt 3 } \right)\left( {\sqrt 2 – \sqrt {2 + \sqrt 3 } } \right)}}{{2 – \left( {2 + \sqrt 3 } \right)}} + \frac{{\left( {2 – \sqrt 3 } \right)\left( {\sqrt 2 + \sqrt {2 – \sqrt 3 } } \right)}}{{2 – \left( {2 – \sqrt 3 } \right)}}\\
    = \frac{{2\sqrt 2 – 2\sqrt {2 + \sqrt 3 } + \sqrt 6 – \sqrt {3\left( {2 + \sqrt 3 } \right)} }}{{ – \sqrt 3 }} + \frac{{2\sqrt 2 + 2\sqrt {2 – \sqrt 3 } – \sqrt 6 – \sqrt {3\left( {2 – \sqrt 3 } \right)} }}{{\sqrt 3 }}\\
    = 2\sqrt 2 + 2\sqrt {2 – \sqrt 3 } – \sqrt 6 – \sqrt {3\left( {2 – \sqrt 3 } \right)} – 2\sqrt 2 + 2\sqrt {2 + \sqrt 3 } – \sqrt 6 – \sqrt {3\left( {2 + \sqrt 3 } \right)} \\
    = 2\sqrt {2 – \sqrt 3 } + 2\sqrt {2 + \sqrt 3 } – \sqrt {3\left( {2 – \sqrt 3 } \right)} – \sqrt {3\left( {2 + \sqrt 3 } \right)} \\
    = 2\left( {\sqrt {2 – \sqrt 3 } + \sqrt {2 + \sqrt 3 } } \right) – \sqrt 3 \left( {\sqrt {2 – \sqrt 3 } + \sqrt {2 + \sqrt 3 } } \right)\\
    = \left( {\sqrt {2 – \sqrt 3 } + \sqrt {2 + \sqrt 3 } } \right)\left( {2 – \sqrt 3 } \right)\\
    = \left( {\frac{{\sqrt {4 – 2\sqrt 3 } }}{{\sqrt 2 }} + \frac{{\sqrt {4 + 2\sqrt 3 } }}{{\sqrt 2 }}} \right)\left( {2 – \sqrt 3 } \right)\\
    = \left( {\frac{{\sqrt 3 – 1 + \sqrt 3 + 1}}{{\sqrt 2 }}} \right)\left( {2 – \sqrt 3 } \right)\\
    = \frac{{2\sqrt 3 }}{{\sqrt 2 }}\left( {2 – \sqrt 3 } \right)\\
    = \sqrt 3 \left( {2 – \sqrt 3 } \right).
    \end{array}\]

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