  Inspired by two entries published in Ramanujan's lost notebook on Page 355,
B. C. Berndt et al.\cite{MR3351542} presented Riesz sum identities for
Ramanujan entries by introducing the twisted divisor sums. Later, S. Kim
\cite{MR3541702} derived analogous results by replacing twisted divisor sums
with twisted sums of divisor functions. Recently, the authors \cite{devika2023}
of the present paper deduced the Cohen-type identities as well as Vorono\"i
summation formulas associated with these twisted sums of divisor functions. The
present paper aims to derive an equivalent version of the results in the
previous paper in terms of identities involving finite sums of trigonometric
functions and the doubly infinite series. As an application, the authors
provide an identity for $r_6(n)$, which is analogous to Hardy's famous result
where $r_6(n)$ denotes the number of representations of natural number $n$ as a
sum of six squares.
