Period of cos(πn/2)·cos(πn/4)

A product (or sum) of discrete-time periodic signals is periodic if each part is periodic; the LCM of the individual periods is then a period (the fundamental period can be smaller, so it is worth checking).

First factor: $\cos(\pi n/2)$, $\omega_1 = \pi/2$:

$$\frac{\omega_1}{2\pi} = \frac{1}{4} \Rightarrow N_1 = 4$$

Second factor: $\cos(\pi n/4)$, $\omega_2 = \pi/4$:

$$\frac{\omega_2}{2\pi} = \frac{1}{8} \Rightarrow N_2 = 8$$

Both ratios are rational, so both factors are periodic and the product is periodic with

$$N = \text{LCM}(4, 8) = 8$$

Check using a product-to-sum identity:

$$\cos\frac{\pi n}{2}\cos\frac{\pi n}{4} = \frac{1}{2}\left[\cos\frac{3\pi n}{4} + \cos\frac{\pi n}{4}\right]$$

$\cos(3\pi n/4)$: $\frac{3}{8} \Rightarrow N = 8$; $\cos(\pi n/4)$: $N = 8$. LCM = 8, which agrees.

Answer: $x[n]$ is periodic with fundamental period $N = 8$.