Determine whether a signal is periodic or not and get fundamental period Here is my signal

Cos(n/2)*cos(pi*n/4)


cos(n/2) has period 4pi and cos(pi*n/4) has period 8

Now, the question is will the signal be periodic for fundamental period 32pi ?
$\cos\left(\dfrac{n}{2}\right)*\cos\left(\pi*\dfrac{n}{4}\right)$

Now finding out period of $\cos\left(\dfrac{n}{2}\right)$
$\cos\left(\dfrac{n}{2}\right)=\cos\left(2*\pi*\dfrac{n}{N1}\right) \implies \dfrac{1}{2}=2*\dfrac{\pi}{N1} \implies N1=4*\pi$

Now finding out period of second function $\cos\left(\pi*\dfrac{n}{4}\right)$
$\cos\left(\pi*\dfrac{n}{4}\right) = \cos\left(2*\pi*\dfrac{n}{N2}\right)\implies \dfrac{\pi}{4}=2*\dfrac{\pi}{N2} \implies N2=8$.


Now for total function period is LCM of $(N1,N2)$ i.e, $\mathrm{LCM}(4*\pi,8) \implies32*\pi$.
Therefore, the period of the function is $\boxed{32*\pi}$.