Integral of Even Powers of Cosine

Let $n \in \mathbb{N}$. [I_n := \int_0^{2\pi} \cos^{2n} (\theta) d\theta = 2 \pi \frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot (2n -1)}{2 \cdot 4 \cdot 6 \cdot \cdot \cdot (2n)}] We begin by relating...