We know that x(t)=cos(t) is a periodic CT signal because it follows the rule, x(t+T)=x(t). It is periodic with a period T=$ 2*pi $
Now by sampling the CT signal at the right frequencies will yield a periodic DT signal. It is noticed that the signal is a DT periodic signal when it is sampled at rate $ 2*pi $,$ pi $.
When you have a signal at a sampling rate of $ 2*pi $ we get a periodic DT signal.
Now by sampling the CT signal at a random sampling rate of 1 we get a non periodic DT signal.
