Cosine after Cosine

Calculus Level 3

Evaluate:

\[\lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ \dots \int _{ 0 }^{ 1 }{ \cos ^{ 2 }{ \{ \frac { \pi }{ 2n } ({ x }_{ 1 }+{ x }_{ 2 }+\dots +{ x }_{ n })\} \quad { dx }_{ 1 }{ dx }_{ 2 }\cdots { dx }_{ n } } } } } } \]

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