  We use the "tridiagonal representation approach" to solve the
time-independent Schr\"odinger equation for the bound states of generalized
versions of the trigonometric and hyperbolic P\"oschl-Teller potentials. These
new solvable potentials do not belong to the conventional class of exactly
solvable problems. The solutions are finite series of square integrable
functions written in terms of the Jacobi polynomial.
