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