$$e_t(x_2,\ldots,x_n) = \sum^{n-t}_{i=1} (-1)^{i+1} \frac{e_{t+i}(x_1,\ldots,x_n)}{x^i_1}\text{ for every } 0\leq t < n.$$
$e_t$ is the $t^{th}$ elementary symmetric polynomial in the variabel $\{x_2\ldots x_n\}$
$e_{t+i}$ is the ${(t+i)}^{th}$ elementary symmetric polynomial in the variabel $\{x_1\ldots x_n\}$.
The only proving method i familiar with is induction, in which i stuck at the inductive step.
Any idea what method should i use to prove this? Please give me a lead where to start.