 In this video, I want to discuss the four so-called product-to-sum identities, and this is a directional statement, product-to-sum, and they get this name, it's not the most clever name, it's like calling a dog a furry quadruped that barks, right? We're just describing what we see here. The idea of the product-to-sum identities is that you have a product of sign or cosine, and there's four possibilities, sine of A times cosine of B, cosine of A times sine of B, cosine of A times cosine of B, and sine of A, sine of B. So it's like you have a sine cosine, a cosine sine, cosine cosine or sine sine with different angles there. The product-to-sum identities will transform these products into sums or in some cases, differences of sines and cosines, which this can be very advantageous to do at times. For example, in calculus, turning a trigonometric product into a trigonometric sum can be a very useful thing, but there's also times as we've been proving identities where we're useful to do the other direction, we'll do that later. The sum-to-product identities is a separate calculation there. So where do these things come from? Well, this is going to be a little bit unorthodox way of proving trigonometric identities, but what we're going to do is completely valid. What we're going to basically do is we're going to combine the angle-sum and angle-difference identities for sine and cosine in a cool way to create these identities. So let's first look at, for example, sine of A times cosine of B. The way we do this is we're going to take the angle-sum identity for sine here, which just as a reminder, sine of A plus B, this is the same thing, of course, as sine of A cosine of B plus cosine of A times sine of B. Okay, and let's try now the angle-difference identity. This is equal to sine of A cosine of B minus cosine of A sine of B, like so. So now we have two equations. What we're going to do is we're going to add these equations together, right? On the left-hand side, you're going to get sine of A plus sine of A minus B, like so. On the right-hand side, what you're going to see is that the cosine A sine of B, they cancel out, and then you're going to double up. You'll get two times sine of A cosine of B, like so. And so if you divide both sides by two, you get exactly this identity right here. Sine of A times cosine of B is equal to one-half sine of A plus B plus sine of A minus B. So by combining the angle-sum and angle-difference identities for sine, you get the following formula. So where do the other formulas come from? Well, let's back up a little bit, right? The only difference here that we're going to do here is actually the difference, right? We're going to take this time sine of A plus B minus sine of A minus B. So the left-hand side, there's really not much to do there, sine of A plus B minus sine of A minus B. On the right-hand side now for subtracting things, we're going to get that sine A cosine B cancels out. Because here you're actually going to get a double negative, so it adds up to be a positive. You end up with two cosine of A sine of B, like so. So divide both sides by two, and we, of course, end up with this identity like so. The other two come about by combining together the angle-sum and angle-difference identities for cosine. So for example, if you take cosine of A plus B, this looks like cosine of A, cosine of B, minus sine of A, sine of B. Compare that with cosine of A minus B. You get cosine of A, cosine of B, plus sine of A, sine of B, like so. In which case, then, if we were to add these together, well, then the left-hand side is going to give you cosine of A plus B minus cosine of A minus B. On the right-hand side, the cosines are going to double up, so you get two cosine A, cosine B. And then the sines cancel out, divide both sides by two. We've now established this identity right here. Cosine of A times cosine of B is equal to one-half cosine of A plus B plus cosine of A minus B. And then back up a little bit. That's rewinding the tape in case you were wondering. If we do this with subtraction, you end up with cosine of A plus B minus cosine of A minus B, like so. And then this time, you're going to see the cosines cancel out because we're subtracting them. This will actually become a negative. So you get a negative there. And so you get negative two sine of A, sine of B. And so then what we do is we divide both sides by negative two. And the negative sign's going to swap the order of this. Or you just put a negative sign out here. It doesn't really matter which one you prefer. But we've now done this identity right here. So we get these four product-to-sum identities. Again, they're all consequences of combining the angle sum and angle difference identities of sine and cosine together. So that's where they come from. When it comes to these identities, they're a lot more complicated. And by complicated, I mean they're more similar, right? When you compare these forms together, there's a lot of similarities. It's like, okay, I get it. There's always a one half that's predictable. You always have an angle sum and an angle difference, A plus B to A minus B. And you can always put the first A plus B first, right? But then sometimes there's a negative on the first or second term. Sometimes it's a plus. Sometimes it's cosine. Sometimes it's a sine. It can get a little bit confusing. So it's very difficult to purely memorize these things because it's so easy because of the similarities that to make a mistake there. So personally, I'm not one to actually require students to memorize the product of some formulas. If your instructor does require you to memorize these identities, you have my pity. Because honestly, they're useful identities, but they're only useful in a particular niche, right? They don't come up as often as the other ones. As I've taught in previous videos here, when you're trying to work with trigonometric identities, look for the horse before you look for the zebra. These are totally the zebra identities. These ones are much more exotic, much more rare in their use. You should know about them, but I generally don't encourage students to have to memorize them. These are identities worth consulting from a formula sheet if you have access to one when the other identities seem to be less than useful. But let's compare some of these calculations. Let's compare these identities in some calculations. Let's consider cosine of 30 degrees times cosine of 120 degrees. Well, 120 degrees, it's an angle that terminates in the second quadrant, but it references with 60 degrees. So cosine of 30, we'll do that one in just a second, but cosine of 120, this is negative cosine of 60 degrees. Cosine is negative in the second quadrant, and like I said, 120 references to 60 degrees. So we could write this as negative cosine of 30 degrees, cosine of 60 degrees. Now, fortunately, because 60 and 30 are complementary angles, if you forget which one is which, is cosine of 31 half or root 3 over 2. It's root 3 over 2. The good news is if you make the mistake, you'll get it right anyways. This is going to equal negative root 3 over 2 times 1 half. That is to say this is negative the square root of 3 over 4. It's not too hard to compute this thing directly, but you could also treat this cosine of 30 degrees here times cosine of 120 degrees. This is a product of two trigonometric functions, cosine times cosine, for which we can come back to our formula here, cosine A times cosine B. This equals 1 half the sum of angles of cosine plus the difference of angles of cosine. So let's go with that there. This is equal to 1 half cosine of the sum of their angles, so 30 plus 120, like so. And then we're going to add to it cosine of their difference, 30 minus 120. Now, because of the symmetries of sines and cosines, you might be like, well, what if I swap the roll here? What if 120 is A angle A and 30 is angle B? It doesn't matter. The symmetry here will actually correct itself, and you'll see that in just a second. So we're going to get 1 half of cosine of 150 degrees, and then the other one's going to give us a negative 90 degrees, like so. Now, cosine is an even function. So cosine of negative 90 is actually the same thing as cosine of 90, because again, cosine is even in that regard. Cosine of 150 degrees, if you're trying to do this without a calculator, the idea is to remember that our 150 terminates in the second quadrant and it references to 30 degrees. So this will give us negative cosine of 30 degrees. Cosine of 90 degrees, of course, is equal to zero. So I'm just going to drop it off the face of the earth at this moment. Cosine of 30 degrees, like we saw before, was negative root 3 over 2, and so you combine that together, you get negative root 3 over 4 as well. So this is just an example of using the product to some identities here. This doesn't prove them. We proved them on the previous slide right here, but you do see that these things are in fact equal to each other. So these surprisingly strange zebra-like identities do, in fact, they're valid that these things are equal to each other. Now, clearly the first approach seemed a lot easier than the second approach. This is what I mean by the zebras, right? You don't use the product to some identities to make calculations easier in this regard, but they can be useful. Certainly, let me show you such a use. Consider the expression 10 times cosine of 5x times sine of 3x. We're going to turn this product with some or a difference, you know, a difference is just a negative sum there. How do we do that? Well, ignoring the 10, the coefficient of 10, we have a cosine of 5x. We have a sine of 3x, all right? And so this is when you consult your identities, right? It's like, okay, I have a product of cosines and sines. There's one of the product to some identities that seems to apply right here. So you go back to your formula sheet, which it's not on the screen right now, but you would look it up. It's like, okay, cosine, I need a cosine of A times a sine of B. Oh, and then you look on the formula and you're going to see cosine of A sine of B. This is one half sine of A plus B minus sine of A minus B. So that's our product of some identity that's appropriate here. So you can use it. And so you're going to get 10. Again, Ken's just going to stick around for a while. You get one half sine of the angle sums. So sine of 5x plus 3x. And then you're going to get minus sine here of 5x minus 3x. Let's make that actually look like a 3. And then simplify the angles. 10 times one half, of course, is 5. You're going to get sine of 8x minus sine of 2x, like so. And while double angle identities could be used here without knowing what the next step is, we're just going to kind of leave it alone. But what we're able to do is we're able to turn the product into a sum. And that's where these identities get the name. So if ever you're in a situation where you have a product of trig functions, but you'd rather have a sum or difference of trigometric functions, you can use the product to some identities to help you out here. And so, for example, in calculus, there's going to be situations where you want to integrate this bad boy, which is like, yikes, that's a really hard integral. But on the other hand, if you do this thing right here, this is actually a pretty easy integral. What is an integral? Well, we'll worry about that some other day, all right? But the point is, trigometric identities allow you to switch from one form to another. And if the other form is more convenient than the first form, then make that transition and use the new form, of course. Now, I use a calculus oftentimes as a motivating example, even though this is not a video for a calculus course. That's because like the karate kid, we're still learning wax on, wax off, you know, paint the fence to sand the deck. We're learning all these techniques for calculus out of context. But I promise you that in the right moment when the Cobra Kai comes to kick your butt, you'll be prepared because we've been practicing these trigometric skills right now.