\(\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}\)\(=\frac{4+2\sqrt{3}}{\sqrt{4}+\sqrt{4+2\sqrt{3}}}+\frac{4-2\sqrt{3}}{\sqrt{4}-\sqrt{4-2\sqrt{3}}}\)
\(=\frac{4+2\sqrt{3}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{4-2\sqrt{3}}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\)\(=\frac{4+2\sqrt{3}}{2+\sqrt{3}+1}+\frac{4-2\sqrt{3}}{2-\sqrt{3}+1}\)
\(=\frac{\left(\sqrt{3}+1\right)^2}{3+\sqrt{3}}+\frac{\left(\sqrt{3}-1\right)^2}{3-\sqrt{3}}\)
\(=\frac{\left(\sqrt{3}+1\right)^2}{\sqrt{3}\left(\sqrt{3}+1\right)}+\frac{\left(\sqrt{3}-1\right)^2}{\sqrt{3}\left(\sqrt{3}-1\right)}=\frac{\sqrt{3}+1}{\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}}\)
\(=\frac{2\sqrt{3}}{\sqrt{3}}=2\)
