$\int_0^{2\pi}\cos^{2n}\theta\,d\theta$ via a contour integral

Compute \(\int _ { | z | = 1 } \left( z + { \frac { 1 } { z } } \right) ^ { 2 n } { \frac { d z } { z } }\) and use it to show that

\[ \int _ { 0 } ^ { 2 \pi } \cos ^ { 2 n } \theta d \theta = 2 \pi { \frac { 1 \cdot 3 \cdot 5 \cdot \cdot \cdot ( 2 n - 1 ) } { 2 \cdot 4 \cdot 6 \cdot \cdot \cdot ( 2 n ) } } . \]