Is $\exp(-2\sin^2t)$ a characteristic function? Is $\exp(-2\sin^2t)$ the characteristic function of some random variable?

I'll consider a generalization. First, observe that 
$$\phi(t):=\exp(-2 z\sin^2 t)=e^{-z}\exp(z \cos 2t)$$ is a $\pi$-periodic function and so has a Fourier series. (The question above is the case of $z=1$.) To be explicit, recall the Jacobi-Anger expansion of $e^{z \cos t}$ (see equation 10.35.3 in the DLMF) to write $$\phi(t)=\sum\limits_{k=-\infty}^\infty I_k(z)e^{-z}\,e^{2i k t}.$$ Since the complex exponential $e^{in t}$ is the Fourier transform of $\delta(x-n)$, we conclude that $\phi(t)$ is indeed the CF of a discrete random variable taking values at even integers. At this point, working out the particular probabilities (and checking normalization) is a simple exercise which I'll leave to the reader.
