The Simplified value of $$\frac{{\sqrt 6 + 2}}{{\sqrt 2 + \sqrt {2 + \sqrt 3 } }} - \frac{{\sqrt 6 - 2}}{{\sqrt 2 - \sqrt {2 - \sqrt 3 } }} - \frac{{2\sqrt 2 }}{{2 + \sqrt 2 }}$$

Solution(By Examveda Team)

$$\eqalign{ & \frac{{\sqrt 6 + 2}}{{\sqrt 2 + \sqrt {2 + \sqrt 3 } }} - \frac{{\sqrt 6 - 2}}{{\sqrt 2 - \sqrt {2 - \sqrt 3 } }} - \frac{{2\sqrt 2 }}{{2 + \sqrt 2 }} \cr & \Rightarrow \frac{{\sqrt 6 + 2}}{{\sqrt 2 + \frac{{\sqrt 3 + 1}}{{\sqrt 2 }}}} - \frac{{\sqrt 6 - 2}}{{\sqrt 2 - \frac{{\sqrt 3 - 1}}{{\sqrt 2 }}}} - \frac{2}{{\sqrt 2 + 1}} \cr & \Rightarrow \frac{{\left( {\sqrt 6 + 2} \right)\sqrt 2 }}{{2 + \sqrt 3 + 1}} - \frac{{\left( {\sqrt 6 - 2} \right)\sqrt 2 }}{{2 - \sqrt 3 + 1}} - \frac{2}{{\sqrt 2 + 1}} \cr & \Rightarrow \frac{{\sqrt 2 }}{{\sqrt 3 }}\left[ {\frac{{\sqrt 6 + 2}}{{\left( {\sqrt 3 + 1} \right)}} - \frac{{\sqrt 6 - 2}}{{\left( {\sqrt 3 - 1} \right)}}} \right] - \frac{2}{{\sqrt 2 + 1}} \times \frac{{\sqrt 2 - 1}}{{\sqrt 2 - 1}} \cr & \Rightarrow \frac{{\sqrt 2 }}{{\sqrt 3 }}\left[ {\frac{{\sqrt {18} + 2\sqrt 3 - \sqrt 6 - 2 - \sqrt {18} + 2\sqrt 3 - \sqrt 6 + 2}}{{3 - 1}}} \right] - \frac{{2\left( {\sqrt 2 - 1} \right)}}{{2 - 1}} \cr & \Rightarrow \frac{{\sqrt 2 }}{{\sqrt 3 }}\left[ {\frac{{2\left( { - \sqrt 6 + 2\sqrt 3 } \right)}}{2}} \right] - 2\left( {\sqrt 2 - 1} \right) \cr & \Rightarrow - \sqrt 3 \times \sqrt 2 \times \frac{{\sqrt 2 }}{{\sqrt 3 }} + 2\sqrt 3 \times \frac{{\sqrt 2 }}{{\sqrt 3 }} - 2\left( {\sqrt 2 - 1} \right) \cr & \Rightarrow - 2 + 2\sqrt 2 - 2\sqrt 2 + 2 \cr & \Rightarrow 0 \cr} $$