  We describe the trace representations of two families of binary sequences
derived from Fermat quotients modulo an odd prime $p$ (one is the binary
threshold sequences, the other is the Legendre-Fermat quotient sequences) via
determining the defining pairs of all binary characteristic sequences of
cosets, which coincide with the sets of pre-images modulo $p^2$ of each fixed
value of Fermat quotients. From the defining pairs, we can obtain an earlier
result of linear complexity for the binary threshold sequences and a new result
of linear complexity for the Legendre-Fermat quotient sequences under the
assumption of $2^{p-1}\not\equiv 1 \bmod {p^2}$.
