To consider the origin of a pseudo-gap and a superconducting gap found in the high-Tc cuprates, the momentum dependence of the singlet gap parameter and the superconductivity correlation function are evaluated in the t-J model by using an optimization variational Monte Carlo method. In the underdoped regime, the singlet gap is significantly modified from the simple dx2-y2-wave gap (∞ cos kx - cos ky) by the contributions of long-range pairings. Its angular dependence along the Fermi surface is qualitatively consistent with those experimentally observed in both hole- and electron-doped cuprates. This singlet gap will correspond to the pseudo-gap and its doping dependence agrees with that of the pseudo-gap. On the other hand, the superconductivity correlation function is dominant in the nearest-neighbor pairing and its Fourier transform preserves the original simple dx2-y2-wave form. We argue that this superconductivity correlation function is closely related to the coherent superconductivity gap appearing below Tc in the 'Fermi arc' region. Its doping dependence is also consistent with the recent experimental observations.