Property of characteristic function

Question

I have a question about a property of the characteristic function. I know that for each random variable it holds that \begin{equation} \ |\Psi_{X}(t)| \leq 1 \qquad \forall t \in \mathbb{R}, \end{equation} where $\Psi_{X}(t)$ is the characteristic function of $X$. My question is, which random variable satisfies \begin{equation} \ |\Psi_{X}(t)| = 1, \end{equation} for a $t \in \mathbb{R}, t \neq 0$.

Answer 1

HINT / informal argument (which I don't know how to make rigorous):

$$|\Psi_{X}(t)| = |\mathbb E (e^{itX})| = 1 \Rightarrow \mathbb E (e^{itX}) = e^{i\theta}\text{ for some } \theta \in \mathbb{R}$$

Intuitively, this should mean $tX = \theta + 2n\pi$ (for some integer $n$) with probability 1, but I don't know how to make this argument rigorous.