We show that the trigonometric Pöschl-Teller potential well problem of the non-relativistic quantum mechanics is equivalent to a certain model of a linear singular oscillator with the position-dependent mass of the form M(x)=(a^2 m_0)⁄((a^2-x^2)), 0≤x≤a. We found an explicit form of the functions and energy of the wave functions and discrete energy spectrum for this model. Wave functions are expressed through the Jacobi polynomials. At the limit when a→∞ equation of the motion, wave functions and energy spectrum of the model correctly reduce to corresponding results of the usual non-relativistic linear singular oscillator with a constant mass m_0.