How this alleged multiple integral identity can be proved?
$$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {
(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}^{2}-s_{2n}^{2})}= $$ 
$$\int\limits_{-\infty }^{\infty }ds_{1}\int\limits_{-\infty}^{s_{1}}ds_{2}\cdots \int\limits_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos { 
(s_{1}^{2}-s_{2}^{2}+s_{3}^{2}-s_{4}^{2}+\cdots +s_{2n-1}^{2}-s_{2n}^{2})}.$$
The context in which this identity arises can be traced through the similar older question http://mathoverflow.net/questions/125237/multiple-integral-american-mathematical-monthly-problem-11621-and-its-generali