Question

Tính A = \(\frac{1+\frac{\sqrt{3}}{2}}{1+\sqrt{1+\frac{\sqrt{3}}{2}}}-\frac{1-\frac{\sqrt{3}}{2}}{1-\sqrt{1-\frac{\sqrt{3}}{2}}}\)

Answer 1

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

\(A=\frac{2+\sqrt{3}}{3+\sqrt{3}}-\frac{2-\sqrt{3}}{3-\sqrt{3}}=\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)-\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\)

\(=\frac{6\sqrt{3}}{9-3}=\sqrt{3}\)