$\mathbb E x_t$ and $Cov(x_t,x_s)$ for $(x_t=\cos(\lambda t+ \phi)|t \in \mathbb Z)$, $\lambda \sim U([-\pi,\pi])$

Question

Consider the stochastic process $(x_t=\cos(\lambda t+ \phi)|t \in \mathbb Z)$

$\phi \in [-\pi,\pi]$ is a real number and $\lambda \sim U([-\pi,\pi])$ a uniformly distributed random variable.

I'm looking for hints how to calculate $\mathbb Ex_t$ and $Cov(x_t,x_s)$.

I thought about using $\cos(a)\cos(b)=1/2(\cos(a+b)+\cos(a-b))$ but I don't know if there are better ways to solve this task.