When learning about differentation, I came along the product rule: $$D(f \cdot g) = f \cdot Dg + g \cdot Df$$ I immediately thought of this rule from trigonometry: $$ \sin( \alpha + \beta ) = \sin(\alpha) \cdot \cos(\beta) + \cos(\alpha) \cdot \sin(\beta)$$ Is there any relation between these two rules? How can this similarity be explained? Has it something to do with geometry?
For those who don't see the relation that I see: if $\alpha$ is the 'normal' function and $\beta$ is the derivative, then you 'get': $$\sin \cdot D\cos + \cos \cdot D\sin$$ And thus if we name sine $f$ and cosine $g$, we 'get': $$f \cdot Dg + g \cdot Df$$
Of course, I write 'get' but I know you can't do this like that, or can you? That's exactly my question: is there any relation between those two rules/equations stated the top? Or is this some weird thought in my brain? Thanks in advance!