- Another set of identities that we want to become aware of, I don't want to say comfortable with yet, but aware of at least are called product-to-sum formulas. And there's four usual ones that we have available to us. I don't think you need to memorize these, but I think you need to be aware of them and also know where you can find them when you have them. 

So what do they mean? How can we use them? Let's begin and look at if I have the sine of 8x times the sine of 3x, I just want to rewrite that in terms of its sum. So it's a product right now. Rewrite it in terms of a sum. 

I notice that sine is my first one right there, so this becomes 1/2 the cosine of alpha, which is an 8x, minus my beta, which is a 3x, minus the cosine of my alpha plus my beta. And then from there, we can go ahead, and it looks like we can collect a little bit of terms a little bit. 

So I'll have a 1/2 times a cosine of 5x minus the cosine of 11x. And then if you want, you can go ahead and multiply that 1/2 all the way through and go from there. So again, just showing an example of how we can look at a product of sines and, in this particular case, move them into a product of some trigonometric functions or trigonometric formulas. 

Another one that I want to look at is sine of 4x times the cosine of x. Again, just another example right here-- that follows with one that I lined it up right next to it. So 1/2 tells me sine of my alpha plus my beta plus the sine of my alpha minus my beta. 

Collect some terms, sine of 4x plus an x is a sine of 5x plus the sine of-- what was that?-- sine of x. So that's just an x. I don't know what in the world I wrote there. Of a 3x. And I'm done. I can't combine these terms, because they're unlike terms. This is a sine of a 5x. That's a sine of a 3x. So they're unlike terms. I can't go any further. And so we just simply took a product and rewrote it as a sum.