Trapped atoms with long interrogation time provide excellent sample for both spectroscopy and quantum information processing (QIP). Potential well of a trap, however, perturbs internal state of atoms as well as holding them. Even in an optical trap, which is least perturbative among various traps, systematic shift and inhomogeneous broadening of a transition seriously limit its utility as a tool in spectroscopy and QIP. We will report some of the tricks developed to overcome this difficulty. For a transition in an optical frequency domain, magic wavelength using three-level system has been demonstrated for cesium and ytterbium atoms. We have proposed magic power for a pulsed optical trap, which can result in a perturbation-free spectroscopy of a transition in an optical frequency domain. The idea is applicable to a general two-level system and we carried out a proof-of-principle experiment using cesium atoms. For a hyperfine transition of a ground-state alkali metal atoms, dual beam optical trap and spin-echo have been tried to overcome the inhomogeneous broadening with a limited success. Recently we proposed the idea of magic polarization, where difference in the scalar polarizabilities of the two hyperfine levels is compensated by the vector polarizabilities. With a proper polarization of the trapping beam, the vector term can be tuned to eliminate the systematic shift and the inhomogeneous broadening of a microwave transition. We will report our progress in the magic polarization experiment using lithium atoms.