= Solution
The zero means of both amplitudes give $\mathbb EX_t=0$. Their unit variances and zero <covariance> give, for arbitrary integer $s,t$,
$$
\mathbb E[X_sX_t]=\cos(\omega_0s)\cos(\omega_0t)+\sin(\omega_0s)\sin(\omega_0t)
=\cos(\omega_0(s-t)).
$$
Thus the <variance> is one and the <covariance> depends only on lag, proving <weak stationarity>. Independence or normality of the amplitudes is unnecessary. Since
$$
\cos(\omega_0k)=\tfrac12e^{ik\omega_0}+\tfrac12e^{-ik\omega_0},
$$
the <spectral measure of a random harmonic oscillation> assigns mass $1/2$ to each of $-\omega_0,\omega_0$. The corresponding right-continuous <time-series spectral distribution> is
$$
\boxed{F(\lambda)=\begin{cases}0,&\lambda<-\omega_0,\\1/2,&-\omega_0\le\lambda<\omega_0,\\1,&\lambda\ge\omega_0.\end{cases}}
$$
These spectral atoms represent a persistent oscillation. There is no ordinary absolutely continuous spectral density for this process.
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