Delta cosh minus delta sinh

Theorem

The following formula holds: $$\cosh^2_p - \sinh^2_p = e_{-\mu p^2},$$ where $\cosh_p$ denotes the $\Delta$-$\cosh_p$ function, $\sinh_p$ denotes the $\Delta$-$\sinh_p$ function, and $e_p$ denotes the $\Delta$-$e_p$ function.