Comparing 2 sinusoids: Discrete case vs Continuous case
Because of the $2\pi$-periodicity of the cosine and the discrete nature of $n$ ($n,k\in \mathbb{N}$):
$$
x[n] = \cos[\Omega_\mathrm{x}] = \cos[(\Omega_\mathrm{y}+2\pi k)n] = \cos[\Omega_\mathrm{y}n+2\pi kn] = \cos[\Omega_\mathrm{y}n] = y[n]
$$
This important "phenomenon" is called aliasing and only occurs for discrete-time signals. The normalized frequency $\Omega$ is therefore usually restricted to the interval $[-\pi, \pi]$.