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## Angle addition identities

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# Proof of the cosine angle addition identity

## Video transcript

Voiceover: In the last video we proved the angle addition formula for sine. You could imagine in this video I would like to prove the angle addition for cosine, or in particular, that the cosine of X plus Y, of X plus Y, is equal to the cosine of X. Cosine of X, cosine of Y, cosine of Y minus, so if we have a plus here we're going to have a minus here, minus sine of X, sine of X, sine of Y. I'm going to use a very similar technique to the way I proved it for sine. I encourage you to pause the video either now or at any time that you get the inspiration to see if you can do this proof on your own. Just like we thought about it for sine, what is the cosine of X plus Y ^in this diagram right over here? ^Well, X plus Y is this angle right over here. ^If we look at the right triangle ADF, cosine of X plus Y. ^Well, cosine is adjacent over hypotenuse segment AF ^over the hypotenuse, or since the hypotenuse ^is just one, AF divided by one is just going to be AF. ^Cosine of X plus Y is just the length of segment. ^It's just the length of segment AF. ^That right over there is equivalent to this right over here. Let me actually write that down; copy and paste. ^Length of segment AF is cosine of X plus Y. ^Let's think about how we can get that. ^The way I'm going to think about it is, ^given the other right triangles we have in this diagram, ^if we could figure out that or AF. Let me write it this way. This thing, which is the same thing as AF, is equal to. Let me write it this way, it's equal to length of segment AB. It's equal to the length of segment AB, ^which is this entire segment right over here, ^minus the length of segment FB. ^Minus this segment right over here. Minus the length of segment FB. Just from the way our angle addition formula looks for cosine, you might guess what's going to be AB and what's going to be FB. If we can prove that AB is equal to this. If we can prove that FB is equal to this, then we're done. Because we know that cosine of X plus Y, ^which is AF, is equal to AB minus FB. If we can prove this, that it's equal to that minus that. Let's think about what these things actually are. What is AB? ^Let's look at right triangle ACB. We know from the previous video that since triangle ADC has a hypotenuse of one, this length is one, that AC is cosine of X. ^What is AB, well, think about it. ^AB is adjacent to the angle that has measure Y. We could say that the cosine, let me do it down here. We could say, since I've already looked at all of that. ^We could say that the cosine of Y, ^cosine of Y is equal to it's adjacent side. ^The length of it's adjacent side. ^That is segment AB over the hypotenuse ^over cosine of X, cosine of X. ^Or multiply both sides by cosine of X. ^We get that segment AB is equal to cosine of X, ^cosine of X times cosine of Y, cosine of Y, ^which is exactly what we set out to prove. We've just proven that AB is indeed the length of segment AB is indeed equal to cosine X, cosine Y. ^This whole thing is equal to cosine X, cosine Y. ^Now we just have to prove that segment FB is equal to sine X, sine Y. ^This looks like a bit of a strange segment right over here. It's not part of any at least right triangle where we know that I've drawn where we know one of the angles. We can see from this diagram ECBF is a rectangle. We use that fact in the proof for the angle addition formula for sine. We'll also use it now because that tells us that FB is the same as EC, is EC. What is EC going to be equal to? Well, we have this angle Y right over here. What is, let's see, this side is opposite the angle Y. We might want to involve sine. We know that sine of Y, sine of Y, I'm looking right over here, is equal to the length of the opposite side, which is the length of EC, which is the length of EC over the hypotenuse, which is sine of X, sine of X. We figured that out from the last video. If this is X, opposite over hypotenuse is sine of X. Well, the opposite is just one, so the opposite is equal to sine of X. Over here multiply both sides by sine of X, and we get what we were looking for. EC is equal to sine of X, sine of X, times sine of Y, times sine of Y. Once again EC was the exact same thing. Has the same length as segment FB. We have just shown that segment FB is equal to sine of X times sine of Y. This is equal to that right over there. Once again, cosine of X plus Y, which is equal to segment AF, is equal to the length of segment AB minus the length of segment FB, which is equal to, we've proven the length of segment AB is cosine X, cosine Y, minus the length of segment FB, which is sine of X, sine of Y, and we are done.