\(=\dfrac{\sqrt{2}}{2+\sqrt{3}+1}+\dfrac{\sqrt{2}}{2-\sqrt{3}+1}\)
\(=\dfrac{\sqrt{2}}{3+\sqrt{3}}+\dfrac{\sqrt{2}}{3-\sqrt{3}}\)
\(=\dfrac{\sqrt{2}\left(\sqrt{3}-1\right)+\sqrt{2}\left(\sqrt{3}+1\right)}{\sqrt{3}\cdot2}\)
\(=\dfrac{2\sqrt{6}}{2\sqrt{3}}=\sqrt{2}\)
\(\dfrac{1}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\dfrac{1}{\sqrt{2}+\sqrt{2-\sqrt{3}}}\)
\(=\dfrac{\sqrt{2}-\sqrt{2+\sqrt{3}}}{\left(\sqrt{2}+\sqrt{2+\sqrt{3}}\right)\left(\sqrt{2}-\sqrt{2+\sqrt{3}}\right)}+\dfrac{\sqrt{2}+\sqrt{2-\sqrt{3}}}{\left(\sqrt{2}-\sqrt{2-\sqrt{3}}\right)\left(\sqrt{2}+\sqrt{2-\sqrt{3}}\right)}\)
\(=\dfrac{\sqrt{2}-\sqrt{2+\sqrt{3}}}{2-\left(2+\sqrt{3}\right)}+\dfrac{\sqrt{2}+\sqrt{2-\sqrt{3}}}{2-\left(2-\sqrt{3}\right)}\)
\(=\dfrac{\sqrt{2+\sqrt{3}}-\sqrt{2}}{\sqrt{3}}+\dfrac{\sqrt{2}+\sqrt{2-\sqrt{3}}}{\sqrt{3}}\)
\(=\dfrac{\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}}{\sqrt{3}}\)
\(=\dfrac{\sqrt{\left(\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{6}}{2}\right)^2}+\sqrt{\left(\dfrac{\sqrt{6}}{2}-\dfrac{\sqrt{2}}{2}\right)^2}}{\sqrt{3}}\)
\(=\dfrac{\left|\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{6}}{2}\right|+\left|\dfrac{\sqrt{6}}{2}-\dfrac{\sqrt{2}}{2}\right|}{\sqrt{3}}\)
\(=\dfrac{\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{6}}{2}+\dfrac{\sqrt{6}}{2}-\dfrac{\sqrt{2}}{2}}{\sqrt{3}}\)
\(=\dfrac{\sqrt{6}}{\sqrt{3}}=\sqrt{2}\)
