 In a previous video, we learned about the product to sum identities. It turns out there are also four sum to product identities. Again, the direction matters here. These are identities that turn sums into products. Maybe you have a trigonometric sum or difference that you would rather have as a trigonometric product. And so we see the four sum to product identities right here on the screen. You can have a sum of sign and sign. You can have a difference of sign and sign. You can have a sum of cosine and cosine. And you could have a difference of cosine and cosine. Notice that in all of these situations, it's the same. They're either both sign or both cosine. There's not an identity that we're going to talk about right now that combines a plus or minus with a sine and a cosine. And the reason why we just have these four is that these sum to product identities are essentially the inverses of the product to sum identities, which we saw on a previous video. And the way we're going to do that is the following. The way we're going to establish this, I should say, is the following. You'll also notice that in these formulas, we're using alpha and beta as opposed to A and B we did before. And we're going to do the following substitution to get these identities. Say that alpha is equal to A plus B and beta is equal to A minus B. In the product to sum identities, there was a lot of A plus B's, a lot of A minus B's because we use the angle sum and angle difference identities to create the product to sum identities. So to reverse that, we're going to consider this A plus B, A minus B. We'll consider those the angles alpha and beta that we're starting with. And so then if you think of this as a system of equations like we did earlier, right? This is something we were doing with those angle sum identities. You could take A plus B is equal to alpha, A minus B, which is equal to beta. Of course, if you add these together, the B's will cancel out and you're going to get a 2A is equal to alpha plus beta like so, which this then gives us that A, let me put it over here, A is equal to alpha plus beta over 2, all right? Then if we do this thing again, that is to say if we take A plus B, which is equal to alpha, and then we subtract from it A minus B, which is equal to beta, this time the A's will cancel out. The B's will double up because you have a double negative. You get B plus B, which is 2B, and then you'll get A minus beta right there. So this tells you that B is equal to alpha minus beta over 2. So then what you do is you take this interpretation of A and this interpretation of B, you put these into the formulas like so, and then you have to divide by 2. So if you look at the first one, this is just now the angle sum, excuse me, the product to sum identity. So this is sine of A times cosine of B is equal to 1 half sine of A plus B plus sine of A minus B like so. So with the change of angles, you can see that the sum of the product identities are just the product to sums. Again, just changing the perspective a little bit. So with that in mind, we get sine of alpha plus sine of beta is equal to 2 sine of alpha plus beta over 2 times cosine of alpha minus beta over 2. We also get sine of alpha minus sine of beta is equal to 2 cosine alpha plus beta over 2 sine of alpha minus beta over 2. You get cosine of alpha plus cosine of beta is equal to 2 cosine of alpha plus beta over 2 times cosine of alpha minus beta over 2. And then finally, cosine of alpha minus cosine of beta is equal to negative 2 sine of alpha plus beta over 2 times sine of alpha minus beta over 2. And again, if you look at all the product of some identities and you make the substitution, switch each A and B with alpha plus beta over 2 and alpha minus beta over 2. And if you take the A plus Bs and A minus Bs and swap those for alpha and betas and then divide by 2, this is exactly where these formulas come from. Take the four product to some identities with these change of angles substitutions, you get the sum to product identities. And what do you use them for? Well, as the name suggests, they turn sums into products. So if you have a sum of trig functions of these forms, you can turn them into products, assuming that product is more useful than what you started with. All right, consider the following example. Let's take cosine of 30 degrees plus cosine of 90 degrees. This is not too hard to compute just directly, right? Cosine of 30 degrees is gonna be root three over two. Cosine of 90 degrees is equal to zero. So we end up with just root three over two, no big deal. But we can also treat this as a, since it's a sum, we could convert this into a product, right? And so we look for the formula that would apply. We need cosine plus cosine of alpha plus cosine of beta. This is equal to this identity right here, two times cosine of alpha plus beta over two times cosine of alpha minus beta over two. So consulting that identity, we could then apply that right here, we get two times, we're gonna get cosine of the sum of the two angles, 30 plus 90 over two. Then we're gonna take cosine of the difference here, 30 minus 90 over two. Simplifying this as we can, 30 plus 90 degrees, of course is gonna be 120 degrees, this is over two. And then when we take the difference, we're gonna get negative 60 over two. Taking half of the angles, we get cosine of 60 degrees. We're then gonna take cosine, take half the angle negative 30 degrees like so. And so then compute these things. Notice that since cosine's an even function, cosine of negative 30 is the same thing as cosine of positive 30. So you end up with two times cosine of 60, which is one half, those are gonna cancel. And then you get cosine of 30 degrees again, which is root three over two, like we saw before. So it's a little more complicated way of computing it, but the point is just to establish that these two calculations do in fact always give you one and the same thing. But like we saw with the product of some identities here, these sum to product identities, they're very much zebra identities. These are not the things we use immediately, we only use them when it becomes clear or obvious why we should be using them. Clearly the calculation was much easier to do directly than with the sum to product identities. Let me give you a setting where it'd be appropriate to use this identity. Let's say we wanna prove that negative tangent of x is equal to cosine of three x minus cosine of x over sine of three x plus sine of x. You'll notice on the right-hand side, you have a three x and you have angle x, but on the right-hand side, the left-hand side, you just have angle x right there. How are you going to combine these together? There's some way you have to combine these angles together to work them out. And so proving this identity, it turns out the sum to product identities are exactly what you wanna use. Let's take the more complicated side, which is the left-hand side there, right? So you're gonna get cosine of three x minus cosine of x over sine of three x plus sine of x, like so. And so noticing that you have a three x of three x and you have an x and an x, this seems and also you have differences in sums. This one really indicates to me that a sum to product identity would be appropriate in this situation. So the first one, we have a cosine minus a cosine. If we consult our list again, which I won't bring it on the screen, but cosine of alpha minus cosine of beta, this becomes negative two sine of alpha plus beta. So we get three x plus x over two. And then we're gonna times that by sine of alpha minus beta over two. So we get three x minus x over two, like so. We'll come back to it. Then in the denominator, we have sine of three x plus sine of x. We need to look for the sum identity there of sine. And again, without showing you on the screen, sine of alpha plus sine of beta, that was equal to two sine of alpha plus beta over two. So we're gonna get three x plus x over two. Notice how these are the same angle, even though they're not simplified yet. And then we're gonna get on the next one, excuse me, cosine of alpha minus beta over two. So three x minus two over two, like so. So notice again, without even simplifying, I already know that these angles are gonna be the same. Notice on the top and bottom, I have a sine of three x plus x over two. Yes, three x plus x is four x over two is a two x. But notice that you have a sine of three x plus x over two over a sine of three x plus x over two. They're gonna cancel out. I don't have to bother simplifying the angle because they're gonna cancel out. Likewise, these twos are gonna cancel out. The remaining sine and the ring cosine don't cancel out. So let's actually work with them. Let's simplify their angle. Three x minus x is a two x. So we get negative sine of two x over two. This is above cosine of two x over two. The twos cancel, of course, that is to say that these twos cancel and these twos cancel. So this gives us a negative sine of x over cosine of x, a negative sine x over cosine of x. And so in the end, you get the negative tangent of x that we're looking for. And so this actually with the left-hand side. I just noticed that I wrote right hand. I wrote left-hand side earlier when I meant to say right-hand side. Sorry about that. Anyhow, we've proven the identity and it turns out that the sum to product identities are exactly the ones that were necessary in this situation. Again, these are very much zebra identities. These are not the first ones we go looking for. These are the ones we end up with. So unless like there's a perfect match, like this one was very much aligned for sum to product identities. The sum to product identities are generally ones you don't memorize. If your instructor makes you memorize the sum to product identities, you have my pity on your soul because they're so similar. It's difficult to memorize them out of context. They're very easy to... I should say it's not too difficult to recognize when you actually need to use them. And then referencing the formula sheet doesn't seem too hideous of a trigonometric students thing to be able to do. In my personal opinion, of course. But with that said, we'll end on that despairing note. That then brings us to the end of our lecture about the sum to product and product to sum identities which we talked about in lecture 20 of our lecture series. This ends chapter six about trigonometric identities. So thanks for watching. If you learned anything in this chapter about trigonometric identities, please give those videos a like, subscribe to the channel. If you wanna see some more videos about trigonometry or other just mathematical videos in general. And as always, if you have any questions, post them in the comments below and I'll be happy to answer them. Bye, everyone. See you next time.