Sum of squares of delta cosine and delta sine

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Proposition: The following formula holds: $$\cos_p^2(t,t_0)+\sin_p^2(t,t_0)=e_{\mu p^2}(t,t_0),$$ where $\cos_p$ denotes the $\Delta$-$\cos_p$ function and $\sin_p$ denotes the $\Delta$-$\sin_p$ function.

Proof: