Is $y(t) = \cos(t) + x(t)$ a time-invariant system?
$y(t-k) = \cos(t-k) + x(t-k)$
But it isn't equal to $\cos(t) + x(t-k)$
So, would it be time-invariant?
You've answered the question yourself already. So, does the response to a shifted input equal the shifted response to the original (non-shifted) input? If so, then the system is time-invariant. Otherwise, it's a time-varying system.