Jacob's ladders, interactions between $\zeta $-oscillating systems, and a $\zeta $-analogue of an elementary trigonometric identity

In our previous papers, within the theory of the Riemann zeta-function we have introduced the following notions: Jacob's ladders, oscillating systems, $\zeta $-factorization, metamorphoses, etc. In this paper we obtain a $\zeta $-analogue of an elementary trigonometric identity and other interactions between oscillating systems.