  The exact solution of the one-dimensional Schr\"odinger equation with
symmetric trigonometric double-well potential (DWP) is obtained via angular
oblate spheroidal function. The results of stringent analytic calculation for
the ground state splitting of ring-puckering vibration in the 1,3-dioxole (as
an example of the case when the ground state tunneling doublet is well below
the potential barrier top) and 2,3-dihydrofuran (as an example of the case when
the ground state tunneling doublet is close to the potential barrier top) are
compared with several variants of approximate semiclassical (WKB) ones. This
enables us to verify the accuracy of various WKB formulas suggested in the
literature: 1. ordinary WKB, i.e., the formula from the Landau and Lifshitz
textbook; 2. Garg's formula; 3. instanton approach. We show that for the former
case all three variants of WKB provide good accuracy while for the latter one
they are very inaccurate. The results obtained provide a new theoretical tool
for describing relevant experimental data on IR spectroscopy of ring-puckering
vibrations.
