There are many structures in the cotangent bundle. These include the complete and horizontal lifts of the $F_{\lambda }(7,1)$-structure. The $F_{\lambda }(7,1)$-structure was first extended in $M^{n}$ to $T^{\ast }(M^{n})$ by Das, Nivas, and Pathak. Later, the horizontal and complete lift of the $F_{a}(K,1)$-structure in the tangent bundle was given by Prasad and Chauhan. This paper consists of two main sections. In the first part, we find the integrability conditions by calculating Nijenhuis tensors of the complete lifts of the $F_{\lambda }(7,1)$-structure. Later, we get the results of the Tachibana operators applied to vector and covector fields according to the complete lifts of the $% F_{\lambda }(7,1)$-structure in the cotangent bundle $T^{\ast }(M_{n})$. Finally, we study the purity conditions of the Sasakian metric with respect to the complete lifts of the $F_{\lambda }(7,1)$-structure. In the second part, all results obtained in the first section are obtained according to the horizontal lifts of the $F_{\lambda }(7,1)$-structure in cotangent bundle $T^{\ast }(M_{n})$.