Derivative of alternative delta cosine

Theorem

Let $\mathbb{T}$ be a time scale. The following formula holds: $$\mathrm{c}_{pq}^{\Delta}(t,s;\mathbb{T})=p(t)\mathrm{c}_{pq}(t,s;\mathbb{T})-q(t)\mathrm{s}_{pq}(t,s;\mathbb{T}),$$ where $\mathrm{c}_{pq}$ denotes the alternative delta cosine and $\mathrm{s}_{pq}$ denotes the alternative delta sine.