  Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category
with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study
complete cohomology of objects in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ by
applying $\xi$-projective resolutions and $\xi$-injective coresolutions
constructed in $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Vanishing of complete
cohomology detects objects with finite $\xi$-projective dimension and finite
$\xi$-injective dimension. As a consequence, we obtain some criteria for the
validity of the Wakamatsu Tilting Conjecture and give a necessary and
sufficient condition for a virtually Gorenstein algebra to be Gorenstein.
Moreover, we give a general technique for computing complete cohomology of
objects with finite $\xi$-$\mathcal{G}$projective dimension. As an application,
the relationships between $\xi$-projective dimensions and
$\xi$-$\mathcal{G}$projective dimensions for objects in
$(\mathcal{C},\mathbb{E},\mathfrak{s})$ are given.
