Covariance function of a Gaussian Process, sin and cos

Question

Let $X(t) = \varepsilon_1 \cos(t)+ \varepsilon_2 \sin(t)$ where $\varepsilon_1, \varepsilon_2 ∼ N(0, 1)$ iid.

Find the covariance function $C_X(s, t)$.

I know that E(X) = 0 so Cov(X(s),X(t)) = E(X(s)X(t)) but I'm not sure how to proceed after

Answer 1

Note that$$\begin{align}C_X&=E((\varepsilon_1\cos s+\varepsilon_2\sin s)(\varepsilon_1\cos t+\varepsilon_2\sin t))\\&=E(\varepsilon_1^2\cos s\cos t+\varepsilon_2^2\sin s\sin t+\varepsilon_1\varepsilon_2(\cos s\sin t+\sin s\cos t)).\end{align}$$Since $E\varepsilon_1^2=E\varepsilon_2^2=1$ and $E\varepsilon_1\varepsilon_2=0$, this simplifies to $\cos s\cos t+\sin s\sin t=\cos(s-t)$.