how do you prove determinant expand by the Leibniz formula or the Laplace formula is the same?
It depends what your definition of the determinant is. Do you define it as the unique alternating multilinear form $\displaystyle K:\mathbb{R}^n\times\cdots\times\mathbb{R}^n\to \mathbb{R}$ for which $\displaystyle \det(I)=1$? If so, see my blog post here. If not, give us the definition you're using.
my definition of determinant is in here:
Leibniz formula for determinants - Wikipedia, the free encyclopedia