\(A=\dfrac{\sqrt{2}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}+\dfrac{\sqrt{2}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}\)
\(=\dfrac{\sqrt{2}\cdot\sqrt{2}}{2-\sqrt{4-2\sqrt{3}}}+\dfrac{\sqrt{2}\cdot\sqrt{2}}{2+\sqrt{4+2\sqrt{3}}}\)
\(=\dfrac{2}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}+\dfrac{2}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}\)
\(=\dfrac{2}{2-\sqrt{3}+1}+\dfrac{2}{2+\sqrt{3}+1}\)
\(=\dfrac{2}{3-\sqrt{3}}+\dfrac{2}{3+\sqrt{3}}\)
\(=\dfrac{2\left(3+\sqrt{3}\right)+2\left(3-\sqrt{3}\right)}{6}=\dfrac{12}{6}=2\)
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