The Klein-Gordon and the Dirac equations with vector and scalar potentials are investigated under a more general condition, Vv≤V s+constant. These isospectral problems are solved in the case of squared trigonometric potential functions and bound states for either particles or antiparticles are found. The eigenvalues and eigenfunctions are discussed in some detail. It is revealed that a spin-0 particle is better localized than a spin-1/2 particle when they have the same mass and are subjected to the same potentials. © 2007 The Royal Swedish Academy of Sciences.