\(E=\frac{2+\sqrt{3}}{2+\sqrt{\left(\sqrt{3}\right)^2+2\sqrt{3}+1}}+\frac{2-\sqrt{3}}{2-\sqrt{\left(\sqrt{3}\right)^2-2\sqrt{3}+1}}\)
\(E=\frac{2+\sqrt{3}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{2-\sqrt{3}}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(E=\frac{2+\sqrt{3}}{2+\sqrt{3}+1}+\frac{2-\sqrt{3}}{2-\left(\sqrt{3}-1\right)}\)
\(E=\frac{2+\sqrt{3}}{3+\sqrt{3}}+\frac{2-\sqrt{3}}{3-\sqrt{3}}\)
\(E=\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\)
\(E=\frac{6-2\sqrt{3}+3\sqrt{3}-3+6+2\sqrt{3}-3\sqrt{3}-3}{9-3}\)
\(E=\frac{6+6-3-3}{6}=\frac{6}{6}=1\)
VẬY \(E=1\)
\(E=\frac{2+\sqrt{3}}{2+\sqrt{4+2\sqrt{3}}}+\frac{2-\sqrt{3}}{2-\sqrt{4-2\sqrt{3}}}\)
\(E=\frac{2+\sqrt{3}}{2+\sqrt{3+2\sqrt{3}+1}}+\frac{2-\sqrt{3}}{2-\sqrt{3-2\sqrt{3}+1}}\)
\(E=\frac{2+\sqrt{3}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{2-\sqrt{3}}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(E=\frac{2+\sqrt{3}}{2+\sqrt{3}+1}+\frac{2-\sqrt{3}}{2-\sqrt{3}+1}\)
\(E=\frac{2+\sqrt{3}}{3+\sqrt{3}}+\frac{2-\sqrt{3}}{3-\sqrt{3}}\)
\(E=\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\)
\(E=\frac{6+\sqrt{3}-3+6-\sqrt{3}-3}{9-3}\)
\(E=\frac{6}{6}=1\)
\(E=\frac{2+\sqrt{3}}{2+\sqrt{4+2\sqrt{3}}}+\frac{2-\sqrt{3}}{2-\sqrt{4-2\sqrt{3}}}\)
\(E=\frac{\left(2+\sqrt{3}\right)\left(2-\sqrt{\left(\sqrt{3}-1\right)^2}\right)+\left(2-\sqrt{3}\right)\left(2+\sqrt{\left(\sqrt{3}+1\right)^2}\right)}{\left(2+\sqrt{\left(\sqrt{3}+1\right)^2}\right)\left(2-\sqrt{\left(\sqrt{3}-1\right)^2}\right)}\)
\(E=\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}+1\right)+\left(2-\sqrt{3}\right)\left(2-\sqrt{3}+1\right)}{\left(2+\sqrt{3}+1\right)\left(2-\sqrt{3}+1\right)}\)
\(E=\frac{\left(2+\sqrt{3}\right)\left(3+\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(3-\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\)
\(E=\frac{6+2\sqrt{3}+3\sqrt{3}+3+6-2\sqrt{3}-3\sqrt{3}+3}{9-3}\)
\(E=\frac{12+6}{6}=3\)
=\(\frac{2+\sqrt{3}}{2+\left(\sqrt{3+2\sqrt{3}+1}\right)}+\frac{2-\sqrt{3}}{2-\left(\sqrt{3-2\sqrt{3}+1}\right)}\)
=\(\frac{2+\sqrt{3}}{2+\left(\sqrt{3}+1\right)^2}+\frac{2-\sqrt{3}}{2-\left(\sqrt{3}-1\right)^2}\)
=\(\frac{2+\sqrt{3}}{2+|\sqrt{3}+1|}+\frac{2-\sqrt{3}}{2-|\sqrt{3}-1|}\)
=\(\frac{2+\sqrt{3}}{2+\sqrt{3}+1}+\frac{2-\sqrt{3}}{2-\sqrt{3}+1}\)
=\(\frac{2+\sqrt{3}}{3+\sqrt{3}}+\frac{2-\sqrt{3}}{3-\sqrt{3}}\)
=\(\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\)
=\(\frac{6-2\sqrt{3}+3\sqrt{3}-3+6+2\sqrt{3}-3\sqrt{3}-3}{6}\)
=\(\frac{6}{6}\)
= \(1\)
