TeX source:
\begin{array}{rcl} \sqrt{-1+\dfrac{2\sqrt{3}}{3}}\sqrt{-1+\dfrac{2\sqrt{3}}{3}+1}-1 &=& -\sqrt{-1+\dfrac{2\sqrt{3}}{3}}\sqrt{-1+\dfrac{2\sqrt{3}}{3}+2} \\\\\sqrt{-1+\dfrac{2\sqrt{3}}{3}}\sqrt{\dfrac{2\sqrt{3}}{3}}-1 &=& -\sqrt{-1+\dfrac{2\sqrt{3}}{3}}\sqrt{1+\dfrac{2\sqrt{3}}{3}} \\\\\sqrt{\left(-1+\dfrac{2\sqrt{3}}{3}\right)\dfrac{2\sqrt{3}}{3}}-1 &=& -\sqrt{\left(-1+\dfrac{2\sqrt{3}}{3}\right)\left(1+\dfrac{2\sqrt{3}}{3}\right)} \\\\\sqrt{-\dfrac{2\sqrt{3}}{3}+\dfrac{4\cdot 3}{9}}-1 &=& -\sqrt{-1-\dfrac{2\sqrt{3}}{3}+\dfrac{2\sqrt{3}}{3}+\dfrac{4\cdot 3}{9}} \\\\\sqrt{\dfrac{4-2\sqrt{3}}{3}}-1 &=& -\sqrt{-1+\dfrac{4}{3}} \\\\\dfrac{\sqrt{4-2\sqrt{3}}-\sqrt{3}}{\sqrt{3}} &=& -\sqrt{\dfrac{1}{3}} \\\\\dfrac{\sqrt{3-2\sqrt{3}+1}-\sqrt{3}}{\sqrt{3}} &=& -\dfrac{1}{\sqrt{3}} \\\\\dfrac{\sqrt{\left(\sqrt{3}-1\right)^2}-\sqrt{3}}{\sqrt{3}} &=& -\dfrac{1}{\sqrt{3}} \\\\\dfrac{\sqrt{3}-1-\sqrt{3}}{\sqrt{3}} &=& -\dfrac{1}{\sqrt{3}} \\\\\dfrac{-1}{\sqrt{3}} &=& \dfrac{-1}{\sqrt{3}}\end{array}