Question

\(\dfrac{1+\dfrac{\sqrt{3}}{2}}{1+\sqrt{1+\dfrac{\sqrt{3}}{2}}}+\dfrac{1-\dfrac{\sqrt{3}}{2}}{1-\sqrt{1-\dfrac{\sqrt{3}}{2}}}\)

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Answer 1

\(\dfrac{2\left(1+\dfrac{\sqrt{3}}{2}\right)}{2\left(1+\sqrt{1+\dfrac{\sqrt{3}}{2}}\right)}\) + \(\dfrac{2\left(1-\dfrac{\sqrt{3}}{2}\right)}{2\left(1-\sqrt{1-\dfrac{\sqrt{3}}{2}}\right)}\)

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

= \(\dfrac{3+\sqrt{3}+3-\sqrt{3}}{9-3}\) = \(\dfrac{6}{6}\) = \(1\)