Question for covariance stationary process

Question

Given a random variable Y with characteristic function C(w) = E[exp(iwy)] . Let the random process X(t) be defined as X(t)=cos(wt+y). Show that the process X(t) is covariance stationary if C(1)-C(2)=0. I have tried expanding the characteristic function in the trigonometric form and using the result to prove stationarity but I'm still unable to solve it. Any hint will be helpful.

Answer 1

Hint

$$E[X(t)]{=E[\cos(wt+y)]\\=E\left[{e^{iwt+iy}+e^{-iwt-iy}\over 2}\right]\\=\text{const}}$$and$$E[X(t_1)X(t_2)]{=E[\cos(wt_2+y)\cos(wt_1+y)]\\={1\over 2}E[\cos(w(t_2-t_1))]\\+{1\over2}E[\cos (wt_1+wt_2+2y)]}$$since $E[X(t_1)X(t_2)]$ must be a function of $t_1-t_2$ so $E[\cos (wt_1+wt_2+2y)]=0$. The rest is easy.