  We solve the eigenvalue spectra for two quasi exactly solvable (QES)
Schr\"odinger problems defined by the potentials $V(x;\gamma,\eta) =
4\gamma^{2}\cosh^{4}(x) + V_{1}(\gamma,\eta) \cosh^{2}(x) + \eta \left( \eta-1
\right)\tanh^{2}(x)$ and $ U(x;\gamma,\eta) = -4\gamma^{2}\cos^{4}(x) -
V_{1}(\gamma,\eta)\cos^{2}(x) + \eta \left( \eta-1 \right)\tan^{2}(x)$, found
by the anti-isospectral transformation of the former. We use three methods: a
direct polynomial expansion, which shows the relation between the expansion
order and the shape of the potential function; direct comparison to the
confluent Heun equation (CHE), which has been shown to provide only part of the
spectrum in different quantum mechanics problems, and the use of Lie algebras,
which has been proven to reveal hidden algebraic structures of this kind of
spectral problems.
