Hello, wonderful mathematics people. This is Anna Cox from Kellogg Community College developing product to some formulas. We're going to start with our cosine alpha minus beta formula that says cosine alpha, cosine beta plus sine alpha, sine beta. And we're also going to have our cosine alpha plus beta formula, cosine alpha, cosine beta, minus sine alpha, sine beta. If we took these two formulas and we added the formulas, we'd end up with cosine alpha minus beta plus cosine alpha plus beta equaling 2 cosine alpha cosine betas because the sine would cancel. Then if we wanted it to be a product to some formula, we want this cosine alpha, cosine beta. So we want to get rid of the two. So we're just going to divide each side by 1/2. So 1/2 cosine alpha minus beta plus cosine alpha plus beta equal cosine alpha cosine beta. That's one of our product to sum formulas. If we had done the same thing, but this time we subtracted, we'd get cosine alpha minus beta minus cosine alpha plus beta. This time the cosine alpha cosine betas are going to cancel because we're subtracting, and that's going to equal to sine alpha sine beta. To get the sine alpha sine beta by itself, we're going to divide each side by two. So we get 1/2 cosine alpha minus beta minus cosine alpha plus beta, equaling sine alpha sine beta. Now we're going to do the same thing starting with the sine formulas. So if we have sine alpha plus beta and sine alpha minus beta, if we think about adding these, we'd get sine alpha plus beta plus sine alpha minus beta equaling 2 sine alpha cosine beta or 1/2 sine alpha plus beta plus sine alpha minus beta equals sine alpha cosine beta and our last one's going to come from subtracting. If we subtract these, we're going to get sine alpha plus beta minus sine alpha minus beta. This time the sine alpha cosine betas are going to cancel and we get 2 cosine alpha sine beta. To get the cosine alpha sine beta by itself, we're just going to divide by that too. Now these last two are actually very similar. The thing we need to realize is the location of the different angles. So this one was sine alpha, cosine beta, this one's cosine alpha, sine beta. Now we know angles are interchangeable, so we could actually have this as the same equation as long as we paid attention to which is our alpha and which is our beta. So those are the product of sum formulas. We're going to now have sum to product formulas. The sum to product formula says sine alpha plus sine beta equal to sine alpha plus beta, cosine alpha minus beta. If we use the formula we just developed the sine times cosine, we would have the second part portion over here equaling 1/2 by the formula sine of the two angles added together plus sine of the two angles subtracted. So the two times the 1/2 are going to cancel alpha plus beta over 2 plus alpha minus beta over 2 because they have the same denominators. The betas are going to cancel and we're going to get 2A over 2, which we know is just really alpha and we would get sine. And the second one, the alphas are going to cancel and we're going to get 2 beta over 2 or beta. So we've just shown that this is indeed an identity. So sine alpha plus sine, beta equal to sine, alpha plus beta over 2, cosine alpha minus beta over 2. Now some books actually start with the formula 1/2 sine alpha plus beta plus sine alpha minus beta equaling sine alpha, cosine beta. We developed this formula a minute ago. And then what they do is they say, OK, let's let alpha plus beta over 2 equals some new variable. No. Let's let alpha plus beta equals some new variable X. Let's let alpha minus beta equals some new variable Y. If we added these together, we'd get 2A equaling X + y, or alpha would equal X + y / 2. If we took these two and subtracted, we'd get 2 beta equaling X -, y or beta equaling X -, y / 2. And we know variables are just variables. So if we got rid of this half by multiplying it to the other side and we would replace alpha plus beta is just going to be X, alpha minus beta is just going to be Y. Multiply that 1/2 across to get 2:00 on the other side. Instead of alpha, we're going to have X + y / 2. And instead of beta, we're going to have X -, y over 2. And if we remember variables are just variables. This equation and that equation are really the same. So if we use our product to sum formula to show that the sum to product formula is an identity, the two stays there. If we have a sine times a cosine, the formula we just showed was 1/2 sine of the sum of the two angles plus sine of the difference of the two angles. So the two and a half are going to cancel. If we look at this, we end up with 2A over 2 which is just alpha. If we look at the second portion, we end up with -2 beta over 2 or negative beta, and by our even and odd identities we know that Sine's odd, so sine of negative B is negative sine B. Another way to do this is to think about variables or just unknowns. So if I let alpha plus beta equal X, alpha minus beta equal Y, if I added those together, I'd get 2A equaling X + y, or alpha equal X + y / 2. If I had subtracted my alpha plus beta and my alpha minus beta, I'd get 2 beta equaling X -, y all over 2. So then if we look at this formula, we'd have 1/2 instead of actually, let's take the 1/2 to the other side. Instead of alpha plus beta, we're going to have an X. Instead of alpha minus beta, we're going to have AY. We're going to take the two across cosine alpha X + y / 2, sine beta X -, y / 2. If we look this formula and this formula are really the same, just variables have changed. We know we can multiply in any order we want. We look at the next one. If we have cosine alpha plus cosine, beta equal to cosine, alpha plus beta over 2, cosine alpha minus beta over 2. That's a formula for product to sum for that cosine times cosine. So we get 2 * 1/2 cosine of the two angles subtracted plus cosine of the two angles added. The two times the half are going to cancel, we're going to end up with cosine of 2A over 2 or alpha plus. Oh, the alphas are going to cancel there. Alpha minus alpha, beta minus a negative beta is 2, beta over 2 or beta. When we look here, alpha plus alpha is 2A over 2 and the betas cancel. So we had to know our product to sum formula in order to prove the sum to product formula as an identity. The other way to do it back to the alpha plus beta equal X, alpha minus beta equal Y, 2A equal X + y, etcetera. If we multiply to each side by the two, we'd get cosine Y plus cosine X would equal to cosine X + y / 2 times cosine X -, y / 2. Once again, if we look at those two formulas, they really are the same thing, maybe added in different order or multiplied in different order. The last one -2 sine alpha plus beta over 2, sine alpha minus beta over 2. So the -2 * 1/2 is going to leave us a -1 the alpha plus beta over 2 minus alpha minus beta over 2. The alphas are going to cancel. We get 2 beta over 2 or cosine beta minus. Here the betas are going to cancel and we get 2A over 2 or alpha, and when we distribute the negative we get cosine alpha minus cosine beta. The other way to do this, the cosine alpha minus beta is really Y. The cosine alpha plus beta is really X. Multiply the two across sine of X + y / 2, sine X -, y / 2, and if we wanted the X to be first, we could multiply each side of the equation by a -1. And then we come up with the formula we just had from above. So those two are equivalents. Thank you and have a wonderful day.