  Koplienko \cite{Ko} found a trace formula for perturbations of self-adjoint
operators by operators of Hilbert-Schmidt class $\mathcal{B}_2(\mathcal{H})$.
Later, Neidhardt introduced a similar formula in the case of pair of unitaries
$(U,U_0)$ via multiplicative path in \cite{NH}. In 2012, Potapov and Sukochev
\cite{PoSu} obtained a trace formula like the Koplienko trace formula for pairs
of contractions by answering an open question posed by Gesztesy, Pushnitski,
and Simon in \cite[Open Question 11.2]{GePu}. In this article, we supply a new
proof of the Koplienko trace formula in the case of pair of contractions
$(T,T_0)$, where the initial operator $T_0$ is normal, via linear path by
reducing the problem to a finite-dimensional one as in the proof of Krein's
trace formula by Voiculescu \cite{Voi}, Sinha and Mohapatra
\cite{MoSi94,MoSi96}. Consequently, we obtain the Koplienko trace formula for a
class of pairs of contractions using the Sch\"{a}ffer matrix unitary dilation.
Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint
operators and maximal dissipative operators using the Cayley transform. At the
end, we extend the Koplienko-Neidhardt trace formula for a class of pair of
contractions $(T,T_0)$ via multiplicative path.
