  Let $G$ be a connected algebraic group and let $X$ be a smooth projective
$G$-variety. In this paper, we prove a sufficient criterion to determine the
bigness of the tangent bundle $TX$ using the moment map
$\Phi_X^G:T^*X\rightarrow \mathfrak{g}^*$. As an application, the bigness of
the tangent bundles of certain quasi-homogeneous varieties are verified,
including symmetric varieties, horospherical varieties and equivariant
compactifications of commutative linear algebraic groups. Finally, we study in
details the Fano manifolds $X$ with Picard number $1$ which is an equivariant
compactification of a vector group $\mathbb{G}_a^n$. In particular, we will
determine the pseudoeffective cone of $\mathbb{P}(T^*X)$ and show that the
image of the projectivised moment map along the boundary divisor $D$ of $X$ is
projectively equivalent to the dual variety of the VMRT of $X$.
