How to find the Cumulative Distributive Function of a element in a Stochastic Process?

Question

Let $X$ a random variable with distribution Bernoulli with parameter $p$. For every $t\geq0$ the variable $X_t$ is defined as:

$\begin{align} X_t=\begin{cases} cos(\pi t) & X=0 \\sin(\pi t) & X=1\end{cases} \end{align}$

Find the CDF of $X_t$ and $\mathbb{E}(X_t)$

If i use the definition of CDF then (by the law of total probability):

$P(X_t\leq x)=P(cos(\pi t)\leq x)\cdot (1-p)+P(sin(\pi t)\leq x)\cdot p$

But, i'm stuck here.

1. It is not assumed that when $x$ tends to infinity then the CDF tends to $1$?
2. Maybe $X_t$ is distributed similar to a Bernoulli?