The trig product-to-sum identities look very much alike. You have to pay close attention to the subtle differences so that you can apply them correctly. Even though the product looks nice and compact, it’s not always as easy to deal with in calculus computations — the sum or difference of two different angles is preferred.
The first identity has two angles, A and B. When you multiply the sine of one angle times the cosine of the other angle, you end up with one-half the sum of a sum identity and a difference identity. Whew!
This time, multiply the sines of both angles together, and the result equals one-half the difference between a sum identity and a difference identity:
This identity has a mix-and-match feel to it. Two different angles and two different functions are used. There seems to be something for everyone.
The last product-to-sum identity uses the cosines of two angles:
Just in case you think this is hocus-pocus, here’s an example of one of these new identities. Using A = 45 degrees and B = 30 degrees and the identity
Where did you get those values for the sine of 75 and 15 degrees? For 75 degrees, you use the sine of the sum of 45 degrees and 30 degrees. For 15 degrees, you use the sine of the difference between 45 degrees and 30 degrees. Now, simplifying,