Let {X(t), t ∈ (−∞, +∞)} and {Y(t), t ∈ (−∞, +∞)} be two independent, weakly stationary stochastic processes, whose trend functions are identically 0 and which have the same covariance function C(τ)....

Let {X(t), t ∈ (−∞, +∞)} and {Y(t), t ∈ (−∞, +∞)} be two independent, weakly stationary stochastic processes, whose trend functions are identically 0 and which have the same covariance function C(τ).

Prove: The stochastic process {Z(t), t ∈ (−∞, +∞)} with

Z(t) = X(t) cos ωt − Y(t) sin ωt

is weakly stationary.