  We examine dualities of two dimensional conformal field theories by applying
the methods developed in previous works. We first derive the duality between
$SL(2|1)_k/(SL(2)_k \otimes U(1))$ coset and Witten's cigar model or
sine-Liouville theory. The latter two models are
Fateev-Zamolodchikov-Zamolodchikov (FZZ-)dual to each other, hence the relation
of the three models is named FZZ-triality. These results are used to study
correlator correspondences between large $\mathcal{N}=4$ super Liouville theory
and a coset of the form $Y(k_1,k_2)/SL(2)_{k_1 +k_2}$, where $Y(k_1 , k_2)$
consists of two $SL(2|1)_{k_i}$ and free bosons or equivalently two $U(1)$
cosets of $D(2,1;k_i -1)$ at level one. These correspondences are a main result
of this paper. The FZZ-triality acts as a seed of the correspondence, which in
particular implies a hidden $SL(2)_{k'}$ in $SL(2|1)_k$ or $D(2,1 ; k-1)_1$.
The relation of levels is $k' -1 = 1/(k-1)$. We also construct boundary actions
in sine-Liouville theory as another use of the FZZ-triality. Furthermore, we
generalize the FZZ-triality to the case with $SL(n|1)_k/(SL(n)_k \otimes U(1))$
for arbitrary $n>2$.
