  We show that solutions of the Schr\"odinger equation with a symmetric
P\"oschl-Teller potential of a particular form can be expressed in terms of a
closed combination (not series) of trigonometric functions. Using some
properties of the eigenfunctions of the Schr\"odinger equation and their inner
product we determine a new exact representation of the hypergeometric function
with certain values of parameters in terms of a closed combination of
trigonometric functions. We also obtain new results in an explicit closed form
for integrals with the hypergeometric function and with the specific
combination of trigonometric functions.
