I thought I learned all the trigonometric identities but after taking calculus I learned I was wrong. One thing I was missing was the list of product/sum identities. Here's the list. But I was wondering if there's a way to calculate answers without memorizing these equations. In other words, my question is this - why do they exist? How to calculate these in methodical steps? Thank you so much for any help you can give.

Product to Sum Identities