Let $$u\colon \mathbb{R}\times\mathbb{R}^+\to S^1=\mathbb{R}/2\pi\mathbb{Z}.$$
Consider the composition $$\cos~\circ~u.$$ What are the zeros of this composition?
When I got it right, then
$$(x,t)\mapsto u(x,t):=[\varphi]:=\{\varphi+ v: v=2\pi\mathbb{Z}\}$$ for the $\varphi\in S^1$ for which $u(x,t)=\varphi+2\pi k$ for some $k\in\mathbb{Z}$. That is, $u(x,t)$ is an equivalence class.
I am wondering what are the zeros of $\cos(u)$: I think there are at least three possibilities depending on what is the domain of the cosine here: Is it $S^1=\mathbb{R}/2\pi\mathbb{Z}$ or $\mathbb{R}$? Or maybe only $[0,2\pi)$ in the sense that for each equivalence class $[\varphi]$, the cosine is defined by the representative element $\varphi\in (0,2\pi]$, i.e. $\cos([\varphi]):=\cos(\varphi)$?
In this context, I see three possible answers:
(1) The two equivalence classes $$ \left[\frac{\pi}{2}\right]\text{ and }\left[\frac{3\pi}{2}\right] $$ are the zeros of $\cos(u)$, hence it has two zeros.
(2) All the element in these two equivalent classes are the zeros of $\cos(u)$, hence it has infinitely many zeros.
(3) Only the wo representative elements $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ of the two equivalent classes above are the zeros of $\cos(u)$, hence it has two zeros.
I really do not know what is the right answer.
(I don't think I understand what your function $u$ is, but perhaps I don't need to to answer this question.)
The codomain of $u$ is $S^1$, and you're applying $\cos$ to $u(x,t)\in S^1$, so the domain of $\cos$ must contain $S^1$. There's no actual mathematics happening here - this is more or less just required by the "grammar" of how the question's been written.
You're right that $[0, 2\pi)$ is a set of representative elements in $\mathbb{R}$ of $S^1$, but the elements of $S^1$ are equivalence classes. So the zeros of $\cos$ are given by $u = [\pi/2], [3\pi/2]$, as in your answer (1). (But answer (3) is also a perfectly good way of counting the number of zeros and finding representatives of their classes, and it's probably what you'd do in practice.)
Having said which... you've been asked to find the zeros of $\cos\circ\, u$, not $\cos$. That is, you need to find all $(x,t)\in\mathbb{R}\times\mathbb{R}^+$ such that $\cos(u(x,t)) = 0$. You've reduced this to the problem of finding all $(x,t)\in\mathbb{R}\times\mathbb{R}^+$ such that $u(x,t) = [\pi/2]$ or $[3\pi/2]$, but you haven't solved it yet!