Is $A\cos(\omega t+\theta)$ a Gaussian random process?
I'm pretty sure that even algebraically, $Z$ will not be Gaussian.  You're multiplying a Gaussian by a random variable that has a distribution that is effectively the histogram of a (co)sine function.



Let's do it empirically, though, following the example here.

Looking first at the densities, you can see that the density estimate for $Z$ (red) is much more central than that of $A$ (green).



Plotting the qqnorm and qqline plots for both, are below. As you can see, the one for $Z$ does not coincide much with the red line.

First, for $A$:


Then, for $Z$:


Regarding stationarity: the signal $Z$ is not stationary at all because it is cyclostationary.



R Code Below

# 28338
T &lt;- 10000
sigma  &lt;- 1

A &lt;- rnorm(T,0,sigma)
theta &lt;- runif(1,0, 2 * pi)

fs &lt;- 100
t &lt;- (1:T)/fs
omega &lt;- 2* pi * 0.0892987
Z &lt;- A * cos(omega*t + theta)

# First diagram
hist(sin(omega*t))

# Second diagram
dA &lt;- density(A)
dZ &lt;- density(Z)
plot(dZ, col="red", lwd=5)
lines(dA, col="green", lwd=5)

# Third diagram
qqnorm(A)
qqline(A, col = 2)

# Fourth diagram
qqnorm(Z)
qqline(Z, col = 2)