Did Trigonometry just fail?

Algebra Level 4

If

\(a\quad +\quad \frac { 1 }{ a } \quad =\quad 2\cos { 6° } \)

then, find :

\(a^{ 1000 }\quad +\quad \frac { 1 }{ { a }^{ 1000 } } \quad +1\)

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