What is the probability density function of the cosine of a gaussian random variable?

Question

I want to find the probability density function of $Y=\cos(X)$, where $X\sim N(\mu, \sigma^2)$. The answer is known when $X$ is uniformly distributed $U(-\pi, \pi)$ and it is an arcsin pdf, given by, $f_X(x) = {1\over{\pi\sqrt{(1-x^2)}}}$. But when $X$ is Gaussian, there are infinite roots of $Y=\cos(X)$ and the solution looks messy. Any help would be appreciated.