Let $Y$ be a random variable with mean zero and variance $\sigma^2$, and let $c$ be a constant. Let $$X_t = Y\cos(ct)$$ When is the process $X_t$ stationary?
I find $E(X_t) = \cos(ct)E(Y) = 0$
$E(X_tX_{t+\tau}) = E(Y^2\cos(ct)\cos(ct+c\tau)) = \cos(ct)\cos(ct+c\tau)E(Y^2)$
Note that $Var(Y) = E(Y^2)-E(Y)^2 \implies E(Y^2) = Var(Y) = \sigma^2$
Then, using the sum and product formula for cosine, we get $$E(X_tX_{t+\tau})= \frac{\sigma^2}{2}(\cos(2ct+c\tau)+\cos(c\tau)) $$
Now, as I understand it, in order for this process to be stationary, we require $E(X_tX_{t+\tau})$ to be only a function of $\tau$. So is the process only stationary when $c=0$? This doesn't seem to be correct based on the proceeding questions.
Let us observe the situation when $\tau=0$. The expectation of $X_t^2$ should not depend on $t$ hence we should have either $\sigma=0$ or $\cos^2\left(ct\right)=1$ for all $t$. The last condition forces $c$ to be a multiple of $\pi$.
Conversely, we can check that if $\sigma=0$ or $c\in \pi\mathbb Z$ then $\left(X_t\right)_{t\in\mathbb Z}$ is weakly stationary. In the first case $X_t=0$ and in the second, $X_t=\left(-1\right)^t Y$.