Let \(\{X(t), t \in(-\infty,+\infty)\}\) and \(\{Y(t), t \in(-\infty,+\infty)\}\) be two independent, weakly stationary stochastic processes, whose trend functions are identically 0 and which have the same covariance function \(C(\tau)\).

Verify: The stochastic process \(\{Z(t), t \in(-\infty,+\infty)\}\) with

\[Z(t)=X(t) \cos \omega t-Y(t) \sin \omega t\]

is weakly stationary.