Polar to Cartesian transform of a Gaussian RV

Question

Let's say I have a RV, $\Theta$ ~ $N(0, \sigma^2)$. Is this analytically possible to transform this to Cartesian using the transform below:

$ X = cos \Theta $,

I mean, is it possible to calculate the pdf of $X$ analytically? If it is, what is the right way to do it?

I know that if $\Theta$ ~ $U[0,1]$, the transform leads to a Gaussian RV. But, I could not find any information when $\Theta$ itself is Gaussian.

Thanks in advance.