Suppose I have two independent standard normally distributed random variables $X_1$ and $X_2$ and form the linear combination of these to define a random process {$Y_t$} by; $$Y_t=X_1\,\text{cos}(t)+X_2\,\text{sin}(t).$$
Then $Y_t$ is also standard normally distributed.
However, I would like to know how the joint distribution $Y_{t_1,t_2}$ would be distributed. My intuition is that $Y_{t_1}$ and $Y_{t_2}$ are both dependent random variables and so their joint distribution may not be normal. How would I go about forming this joint distribution?
The reason I am asking this is because I want to verify that $Y_t$ is a Gaussian process, and hence want to show that the finite-dimensional distributions are all Gaussian.