I've been trying to solve the determinant of the Vandermonde-matrix by using Laplace's formula on it (without transforming the matrix first).
It worked out quite well, I could almost instantly factor out the result that I was supposed to get, however not without the weird factor $$ \sum_{k=0}^{n} \prod_{j=0\\j\neq k}^{n} \frac{t_k}{t_k-t_j} $$
For the formula to hold, the factor has to be equal to $1$, and I've tried it for small $n$, however I'm not seeing why exactly this identity has to hold.
Therefore:
Why does the identity
$$
\sum_{k=0}^{n} \prod_{j=0\\j\neq k}^{n} \frac{t_k}{t_k-t_j} = 1
$$
hold for abitrary $t_i\in\mathbb C$?