To be IR active, the vibration of a bond must result in a substantial change in dipole moment. Since trans-2-butene is symmetrical, will the C=C stretch show up on IR?
Something tells me no because if the C=C bond vibrates, there will be no net change in the bond dipole moment, unlike in cis-2-butene. Any polarization one way in trans-2-butene is cancelled out by a polarization in the opposite direction in trans-2-butene. Cis-2-butene, however, has a permanent dipole and any vibration only exacerbates this dipole.