Cauchy determinantal formula

Let $x_{1},\dots,x_{n},y_{1},\dots,y_{n}$ be some complex indeterminates. The goal of this exercise is to compute the Cauchy determinant:
$$C_{n}(x_{1},\dots,y_{n}):=\mathrm{det}\left(\left(\frac{1}{x_{i}+y_{j}}\right)_{1\leqslant i,j\leqslant n}\right).$$

1. Show that there exists polynomial $P$ in the variables $x_{1},\dots,x_{n},y_{1},\dots,y_{n}$ such that
   $$C_{n}(x_{1},\dots,y_{n})=\frac{P(x_{1},\dots,y_{n})}{\prod_{1\leqslant i,j\leqslant n}(x_{i}+y_{j})}.$$And express its degree. Recall that the degree of a polynomial of multiple variables is the maximum of the sum of powers of variables in each monomial. For instance $7+x_{1}x_{n}+y_{1}^{3}x_{2}$ is a polynomial of degree $4$ as a sum of monomials of degree $0,2$ and $4$ respectively.

2. Compute $C_{n}$ in the situation where two $x_{i}$ are equal. Do the same if two $y_{j}$ are equals.

3. Deduce hence that there exists a scalar $\alpha$ such that
   $$P_{n}(x_{1},\dots,y_{n})=\alpha\prod_{1\leqslant i<j\leqslant n}(x_{i}-x_{j})(y_{i}-y_{j})$$

4. Fix the variables $x_{2},\dots,x_{n},y_{2},\dots,y_{n}$ and consider $Q(x_{1},y_{1})$ the polynomial of two variables such that $Q(x_{1},y_{1}):=P(x_{1},\dots,y_{n})$. Express its degree and dominant coefficient and deduce that $\alpha=1$.

We finally deduce from this exercice the value of the determinant:
$$\boxed{\mathrm{det}\left(\left(\frac{1}{x_{i}+y_{j}}\right)_{1\leqslant i,j\leqslant n}\right)=\frac{\displaystyle\prod_{1\leqslant i<j\leqslant n}(x_{i}-x_{j})(y_{i}-y_{j})}{\displaystyle\prod_{1\leqslant i,j\leqslant n}(x_{i}+y_{j})}}$$

5. As an application, deduce the value of the Hilbert determinant
   $$\mathrm{det}\left(\left(\frac{1}{i+j-1}\right)_{1\leqslant i,j\leqslant n}\right)=\prod_{k=0}^{n-1}\frac{(k!)^{3}}{(k+n)!}.$$