Cossine of a standard Normal distribution

Question

Let $X \sim \mathcal{N}(0,1) $ have a standard normal distribution. How do I compute $$\mathbb{E}[cos(X)] \text{ ?}$$ I know that $$\mathbb{E}[sin(X)] = 0$$ because $$f(x)=sin(x) \frac{e^{-x^2}}{\sqrt{2 \pi}}$$ is an odd function. Can I use this fact to compute $\mathbb{E}[cos(X)]$?

Answer 1

Since $\Bbb Ee^{itX}=e^{-t^2/2}$, $\Bbb E\cos X=\Re\left.\Bbb Ee^{itX}\right|_{t=1}=e^{-1/2}$.