reference request – Does this identity well-known?

Let $a_1, a_2, …, a_n$ and $x$ are real number, the the identity as follows hold:

$$(a_1-frac{x}{a_2})(a_2-frac{x}{a_{3}})(a_3-frac{x}{a_{4}})…(a_n-frac{x}{a_{1}})=(a_1-frac{x}{a_n})(a_2-frac{x}{a_{1}})(a_3-frac{x}{a_{2}})…(a_n-frac{x}{a_{n-1}})$$

If $n=3$, let $a$, $b$, $c$, $x$ are real number, then:

$$(a-frac{x}{c})(b-frac{x}{a})(c-frac{x}{b})=(a-frac{x}{b})(b-frac{x}{c})(c-frac{x}{a})$$

Question: Does this identity well-known?

I hope that can apply the identity to solve some diophantine-equations