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