#2015_3 Not that easy, huh?

Algebra Level 4

If \({ x }_{ 1 }=\sqrt { 2 } \quad ,{ x }_{ 2 }=\sqrt { 2+\sqrt { 2 } },\) and likewise \[ { x }_{ n }=\sqrt { 2+\sqrt { 2+\sqrt { 2+.......n\ terms } } } \], then find \[2\sin\left( 1+\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +........+\frac { 1 }{ { 2 }^{ 2006 } } \right) 45°\].