Expected value of tangent of a normal random variable If $z\sim N(\mu,\sigma^2)$
What is $E[\tan(z)]$ and $E[\tan^2(z)]$?
Generally, it seems that the expectation does not exist. How about if $z$ is bounded $(0,\pi/2)$?
Update: Theoretically, my random variable can change from 0 to $\pi/2$ (the endpoints are excluded). However, practically, its upper bound is smaller than $\pi/3$. That's why, in the comments below, I've jumped to $\pi/3$.
Update 2: By "practically", I mean this: My random variable is in fact a material property. There is no theoretical limitation that avoids this property to approach $\pi/2$. However, nobody has reported any value higher than $\pi/3$  in practice (in nature).
 A: The behavior of the $\tan$ function as the argument approaches $\pi/2$:

guarantees that unless the density approaches 0 rapidly enough as the argument approaches $\pi/2$, the limit ($\lim_{t\to\pi/2} \int_{0}^{t}\tan(x)f(x)dx$) will not be finite. No normal truncated to $(0,\pi/2)$ can approach 0 in a way that would do that. 
If you take $\mu$ well into the negatives or make $\sigma$ quite small, at first glance it can look like it might converge, but it never will: 

The distribution of $\tan(z)$ will exhibit somewhat similar upper-tail behavior to a Cauchy distribution.
By pulling the upper bound away from $\pi/2$ the integral will converge to a finite quantity and can at least be done numerically ... e.g. for a standard normal and an upper limit of $\pi/3$ the expectation is about 0.521564. But the Wolfram online integrator can't do the indefinite integral, so I can fairly safely assume I won't be able to; numerical calculation, series approximations, or bounds are probably your main options.
However, your willingness in comments to jump from an upper bound of $\pi/2$ to one of $\pi/3$ suggests that you have not got a well-defined problem (if it can be changed so readily to $\pi/3$, why not something else? If that's so easily changed, why not the distribution, or the function of the random variable whose expectation you're taking? Why this expectation? How do you know you have normality?). It may be better to focus your interest on more carefully identifying and defining the underlying problem here -- it may lead to more productive lines of enquiry. 

However, practically, its upper bound is smaller than $π/3$.

Everything depends on precisely what "practically" means here. Again, this suggests the same kind of concern I expressed above about the problem not being clearly defined.
Is there some chance values can exceed $\pi/3$, no matter how small? Is there any chance they can closely approach $\pi/2$, no matter how infinitesimally remote those chances might be? If so, can we really say anything about what the rate of decrease might be at higher values?
Here, for example, is a scaled beta on $(0,\pi/2)$ (though almost none of the probability lies beyond $\pi/3$ and even with many, many observations you might not see any values above $\pi/3$:

The integral for this distribution is finite.
(Yet you'd have a hard time telling that beta distribution apart from a normal with the same mean and variance.)
