Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. The sum and difference formulas, if we give ourselves any angle, let's call this alpha, then a point out here would be cosine alpha, sine alpha. We went over a cosine alpha mount and we went up a sine alpha mount. If we gave ourself another angle, let's call it beta, and we put it right next to alpha, then this point here would be cosine alpha plus beta and sine alpha plus beta. How far over we went, how far up we went. What if we wanted to put a negative beta? If we had a negative beta, it's going to come in the opposite direction and we're going to have a point. We would call it cosine negative beta, sine negative beta. Now based on even and odd identities, cosine of negative beta is really the same thing as cosine beta. Sine negative beta is negative sine beta if we have a .10 on the X axis. If we look at our angles, the distance or the angle covered alpha plus beta. So the distance between this cosine alpha plus beta, sine alpha plus beta back to our 10 should be the same distance as our cosine alpha, sine alpha back to our cosine beta, negative sine beta, because they're the same arc alpha plus beta here, alpha plus beta here. So now we're going to do a lot of algebra. We're going to do the distance formula. So if I want the distance formula, the distance formula says it's the square root of X2 minus X 1 ^2 + y two minus Y 1 ^2. It's based off of the Pythagorean theorem. So if we're going to do the two points cosine alpha plus beta -1 ^2 plus sine alpha plus beta -0 ^2, that's got to equal the square root of cosine alpha minus cosine beta squared plus sine alpha minus negative sine beta squared. OK, lots of algebra involved. Next, I'm going to square each side to get rid of the square roots. If I foil, I'm going to get cosine squared alpha plus beta -2 cosine alpha plus beta plus one plus sine squared alpha plus beta. That's the left side of this equation. The right side, we're going to equal cosine squared alpha -2 cosine alpha, cosine beta plus cosine squared beta plus sine squared alpha +2 sine alpha, sine beta plus sine squared beta from foiling. When we look at this, we're going to then group together cosine squared of an angle plus sine squared of the same angle, because cosine squared plus of an angle plus sine squared of the same angle is really just one. So there's a cosine squared, and there's a sine squared, and here is another cosine squared and another sine squared of the same angles. So now we're going to get 1 -, 2 cosine alpha plus beta plus another one on the other side. We're going to get 1 -, 2 cosine alpha cosine beta, plus another 1 + 2 sine alpha sine beta. So the 1 + 1 on each side is going to be a 2, and if I combine my like terms, those twos are going to cancel. Once I get the twos to cancel, I could think about dividing through by a -2. If I divide through by a -2, that will give me cosine alpha plus beta, equaling cosine alpha, cosine beta minus sine alpha, sine beta. And that's our first sum formula. Cosine alpha plus beta equal cosine alpha, cosine beta minus sine alpha, sine beta. We can come up with the other sum and difference formulas pretty easy from this. Cosine of alpha plus negative beta is an angle also, so if it's true for a positive beta, it should hold true also for a negative beta. So this would turn into a cosine alpha, cosine negative beta, minus sine alpha, sine negative beta. Now cosine alpha plus negative beta we could think of as cosine of alpha minus beta. And with our even and odd identities, cosine of negative beta is really just cosine beta. Sine of negative beta is really negative sine beta. So a negative of a negative would make it a positive sine alpha sine beta. That's our second equation. Now what about if we wanted to do sine, sine of alpha plus beta? Well, we can think of this as cosine using our cofunction identity of π halves minus alpha plus beta. If we regrouped, we could think of that as cosine of π halves minus alpha minus beta. Now we can use the formula we just developed. So we have cosine Pi halves minus alpha times cosine beta plus sine of π halves minus alpha, sine beta. Well, what is cosine of π halves minus alpha? That's our cofunction identity for sine. What is our sine Pi halves minus alpha? That's our cofunction identity for cosine. So sine of alpha plus beta equals sine alpha, cosine beta plus cosine alpha, sine beta. That's one of our new formulas. If we use the concept of sine alpha plus beta with sine alpha, cosine beta plus cosine alpha sine beta. But now we're going to put in a negative angle. So instead of alpha plus beta, how about alpha plus negative beta? Negative beta is an angle, so that's going to equal sine alpha, cosine negative beta plus cosine alpha sine negative beta. So sine of alpha minus beta. Our cofunction or not our cofunction are even odd identities. Cosine of negative beta is really just cosine beta, and sine of negative beta is negative sine beta. So a positive and a negative are going to make it a negative. And that's another one of our sum difference formulas. We have these formulas also for tangent and tangent. We're just going to divide. So if we have tangent of alpha plus beta, that's going to be the same thing as sine alpha plus beta over cosine alpha plus beta. Well, sine is sine alpha, cosine beta plus cosine alpha, sine beta. We're going to divide that by cosine alpha, cosine beta minus sine alpha, sine beta. And that looks kind of confusing and we want to simplify it. So what we're going to do is we're going to take each and every term and we're going to try to make them into tangents. So I'm going to multiply by one over sine cosine alpha, cosine beta, cosine alpha, cosine beta over one over cosine alpha, cosine beta. So what that's going to do is it's going to allow us to have the cosines cancel out and leaving us sines over cosines. So if I do each and every term, sine alpha, cosine beta over cosine alpha, cosine beta, the cosine betas are going to cancel, leaving me a sine alpha over cosine alpha, cosine beta over cosine alpha, cosine beta. The cosines are going to cancel this time, leaving me a sine beta divided by a cosine beta. That's the numerator portion. Now we're going to do the cosine alpha, cosine beta divided by cosine alpha, cosine beta. That's all going to cancel leaving us A1. And then the last term we have sine alpha, sine beta over cosine alpha, cosine beta. So this is all going to simplify into tangent alpha plus tangent beta over 1 minus tangent alpha, tangent beta. So that's what our tangent alpha plus beta looks like. Now our minus we're going to do the same way, except we're going to put in as a negative angle. So if we have tangent of alpha plus negative beta, that's just going to be tangent alpha plus tangent negative beta over one minus tangent alpha tangent negative beta. Well, even an odd symmetry. So the tangent negative beta is going to be negative tangent beta. So we're going to get tangent alpha minus tangent beta over one plus tangent alpha, tangent beta equaling our tangent alpha minus beta formula. So those are our six summon difference formulas. Thank you and have a wonderful day.