The main motive of this paper is to study weakly $\varphi$-Ricci symmetric lightlike hypersurface ($W\varphi RS$) $- LH$ of an indefinite cosymplectic manifold ($M$, $g$) of constant curvature $-1$. In this paper, we acquire a relationship between 1-forms of ($W\varphi RS$) $- LH$ of ($M$, $g$). We study $\eta$-Einstein weakly $\varphi$-Ricci symmetric lightlike hypersurface ($W\varphi RS$) $- LH$ of an indefinite cosymplectic manifold ($M$, $g$). Finally it is shown that the Ricci tensor $S$ of ($M$, $g$) is satisfying cyclic parallel and Codazzi-type properties of Ricci tensor.