  One-dimensional potentials defined by $V^{(S)}(x) =S(S+1) \hbar^2 \pi^2
/[2ma^2\sin^2(\pi x/a)]$ (for integer $S$) arise in the repeated
supersymmetrization of the infinite square well, here defined over the region
$(0,a)$. We review the derivation of this hierarchy of potentials and then use
the methods of supersymmetric quantum mechanics, as well as more familiar
textbook techniques, to derive compact closed-form expressions for the
normalized solutions, $\psi_n^{(S)}(x)$, for all $V^{(S)}(x)$ in terms of
well-known special functions in a pedagogically accessible manner. We also note
how the solutions can be obtained as a special case of a family of
shape-invariant potentials, the trigonometric P\"oschl-Teller potentials, which
can be used to confirm our results. We then suggest additional avenues for
research questions related to, and pedagogical applications of, these
solutions, including the behavior of the corresponding momentum-space wave
functions $\phi_n^{(S)}(p)$ for large $|p|$ and general questions about the
supersymmetric hierarchies of potentials which include an infinite barrier.
