We consider two classes of left $R$-modules, $\mathscr{P}$ and $\mathscr{C}$, such that $\mathscr{P}\subset \mathscr{C}$. If the module $M$ has a $\mathscr{P}$-resolution and a $\mathscr{C}$-resolution then for any module $N$ and $n\geq 0$ we define generalized Tate cohomology modules $\widehat{Ext}_{\mathscr{C},\mathscr{P}}^{n}(M, N)$ and show that we get a long exact sequence connecting these modules and the modules $Ext_{\mathscr{C}}^{n}(M, N)$ and $Ext_{\mathscr{P}}^{n}(M, N)$. When $\mathscr{C}$ is the class of Gorenstein projective modules, $\mathscr{P}$ is the class of projective modules and when $M$ has a complete resolution we show that the modules $\widehat{Ext}_{\mathscr{C},\mathscr{P}}^{n}(M, N)$ for $n\geq 1$ are the usual Tate cohomology modules and prove that our exact sequence gives an exact sequence provided by Avramov and Martsinkovsky. Then we show that there is a dual result. We also prove that over Gorenstein rings Tate cohomology $\widehat{Ext}_{R}^{n}(M, N)$ can be computed using either a complete resolution of $M$ or a complete injective resolution of $N$. And so, using our dual result, we obtain Avramov and Martsinkovsky's exact sequence under hypotheses different from theirs.