Let $X_{t}=\cos(\lambda t+U)$ where $\lambda \in R$ and $U$ is a random variable uniformly distrusted on ($-\pi,\pi$]. Determine the covariance function $\gamma_{X}(t,t+h)=Cov(X_{t},X_{t+1})$. is {$X_{t}$} stationary?
Own work: Ihave been trying to determine the covariance using expected value and the trigonometric addition identity. but don't seem to be getting very far. Thanks for any help
so far i know that the expectation of $EX_{t}=0$ from this the covariance is equal to $E(X_{t}X_{t+h})$ $E(X_{t}X_{t+h})=E(cos(\lambda t +U)cos(\lambda(t+h)+U)$
$=E((\cos(\lambda t)\cos(U)-\sin(\lambda t)\sin(U)(\cos(\lambda (t+h)\cos(U)-\sin(\lambda (t+h))\sin(U)))$
Hint: $$\cos(A) \cos(B) = \frac{1}{2} \left(\cos(A-B) + \cos(A+B)\right) $$