 In this video, I want to talk about the double angle formula for cosine, which it turns out there's actually three versions of the double angle formula for cosine of the first of which we're going to talk about this one right here. But there are some consequences that are often pushed with the double angle formula. So I want to talk about each and every one of those. Let's talk about, again, just the first one right here, cosine of 2a is equal to cosine squared of a minus sine squared of a. And to see that, let's take the left-hand side, which is equal to cosine of 2a, 2a right there. And I'm going to treat this as if it's an angle sum identity, where just I use the same angle twice. So this really, we think of as cosine of a plus a right here. And so then if we use the angle sum identity for cosine, we're going to get cosine of a, cosine, well, normally if we were taking cosine of a plus b, we'd get cosine a, cosine b. But as b is just the angle a used twice, we're going to get cosine a, cosine a. And then we're going to get minus sine of a, sine of, well, again, on the standard form it's cosine a, cosine b minus sine a, sine b. But a and b are the same angle, so we just get angle a twice here. And so then simplifying this, we see we're going to get a cosine squared of a minus sine squared of a. That is the right-hand side. And that establishes the first of the angle sum identities, excuse me, the double angle identities for cosine. So cosine of two a equals cosine squared of a minus sine squared of a. And for me personally, this is the one I remember the most, mostly because it resembles most closely the angle sum identity. So you have two cosines and two sines, right? It also kind of looks like a Pythagorean relationship, it's now it's negative instead of plus. So maybe that makes it easier to remember. Now when it comes to practical uses of the double angle identity, this won't be very useful, but there's two other ones to alternate versions that can be very useful. So let's talk about those for a second. So cosine of two a is equivalent to two times cosine squared of a minus one. It's also equivalent to one minus two times sine squared of a. Why would you have these alternative versions? Well the thing is, when you look at cosine of two a, this one involves both cosine and sine, but you've looked at these alternatives. Cosine of two a could be expressed as just two times cosine squared of a minus one. So you could express cosine of two a only using cosine. You don't need to know what sine is. You can get away with just knowing cosine. And the third one, cosine of two a is equal to one minus two times sine squared of a. Well, what do you do with that? Well, if you only know sine, you can still compute cosine of two a. You don't need to know both cosine and sine. So let's talk about this, the proofs of them, right? So how do you prove cosine of two a is equal to two cosine squared of a minus one? Well, the idea is you're going to utilize the Pythagorean relationship. So notice that cosine squared of a plus sine squared of a is equal to one. So if you solve for sine squared, you get that sine squared of a is equal to one minus cosine squared of a. And we're going to make this substitution into the original formula we have right there. So basically, if you prove that one, the left-hand side is equal to cosine of two a. Well, as we've already established, this is equal to cosine squared a minus sine squared a. And then we're going to replace the sine squared with something equivalent that comes from the Pythagorean relationship. You get one minus cosine squared of a, like so. For which then, if you distribute that negative sine, you end up with cosine squared minus one plus cosine squared. So you see that the cosine squared's double up. So you get two cosine squareds of a minus one. And that's then proving the right-hand side, like so. We can do the same trick here for the third one. So if you want to prove that cosine of two a is equal to one minus two sine squared of a, the idea here is you take your Pythagorean relationship again, cosine squared plus sine squared equals one. This time you solve for cosine. And so you get cosine squared of a equals one minus sine squared. And then you're going to substitute in this for the cosine squared right here. So again, the argument's very similar, very quick. You get that cosine of two a, two a. This is equal to cosine squared of a minus sine squared of a. You're going to replace this time the cosine with one minus sine squared of a. You have a minus sine squared of a. And so combining like terms, you get one minus two sine squared of a. And this is equal to the right-hand side. Why do we go through the details of these proofs all the time? Well, one, as a student in trigonometry, you will be expected to prove trigonometric identities, so it's good to see examples of this. Two, as you're starting to see that these lists of trigonometric identities get longer and longer and longer, there's three versions of cosine of two a. How do you remember all of those? Well, the thing is you don't have to remember all of those. If you remember the original one, cosine of two a is equal to cosine squared minus sine squared, if you remember that one, and you remember the Pythagorean identity, then you can splice the two together to get you these alternative ones if pertain to you, you needed them and you didn't remember them. So this is sort of like desert island trigonometry. What do you need? You know, after you've crashed in the Pacific Ocean, you wash up on an island. The only thing that survived is you and your volleyball. How do you do trigonometry? Well, it's like, oh, I still have the double angle identity. I still have the Pythagorean identity. I can then recreate these other identities in, you know, in a rush if I needed to. So let's look at some calculations involving the double angle identity. Let's suppose that we know sine of a is equal to one over the square root of five. Okay, we want to compute cosine of two a. Well, like we saw in the previous slide, cosine of two a is equal to cosine squared of a minus sine squared of a. So if we know sine squared, excuse me, if we know sine and cosine, then we can compute a double angle. We know sine, but we don't know cosine. And so what we could do is we could try to compute cosine directly, but we don't know the quadrant we're in. So the best we could do is we could get the absolute value of cosine. Now, fortunately, because you're squaring it, you don't need to know whether it's positive or negative. It'll be positive when you square it no matter what. So we could do that. But even better is if we know sine, we don't even need to know cosine whatsoever, because cosine of two a is equal to one minus two sine squared of a. Remember how cosine is a jerk and prefers cosine over other sines, right? That's why the negative's in front of the sine. That thing is inherited with these alternative versions, right? So you notice you have a one minus two sine squared. That's because cosine doesn't like sines, so the negative sign will stay in front of the sines there. So we can compute the value with just sine. We don't need to know cosine. That's a huge save of effort. We don't have to bother computing cosine here. So we get one minus two times one over the square root of five squared. When you square this one over the square root of five, you're just going to get one fifth. So you get one minus two fifths right here. So we can think of one as five fifths, five fifths, take away two fifths. Oh boy, that's going to give us three fifths. And so then cosine of two a is going to equal three fifths. We are able to compute that without using information about cosine. Because we are able to choose the form of cosine of two a that's most convenient for us. Let's try to prove a trigonometric identity involving the double angle. This one actually uses the quadruple angle, right? Cosine of four x here. And so believe it or not, I'm actually going to take the left-hand side to start with. Because what I can see here is that the left-hand side since it's cosine of four x, what I'm going to do is I'm going to treat this as the double, double angle. So notice that cosine of four x is cosine of two, two x. For which then I can apply a double angle identity to that. But which double angle identity do I use? Do I use cosine squared minus sine squared? Do I use two cosine squared minus one or do I use one minus two sine squared? Well, this is where we look at the right-hand side here. Notice the right-hand side only involves cosines. I have a cosine to the fourth. I have a cosine squared of constant. There's no signs. So maybe I should use the double angle identity that only involves cosines. So what this is going to give me is that the double angle identity will be two times cosine squared of something minus one. Now that something is the angle we've doubled. So since I started with four x, which is double two x, the two x is what's going to go inside of here. And so then we have two cosine squared of two x minus one. But you'll notice that the cosine squared of two x, it's still a double angle. And so before I apply the double angle identity, I'm going to rewrite this thing to make it very clear that this square is affecting the entire cosine. And so if we apply the double angle identity to cosine of two x there, we're going to square whatever that substitution is. And again, because the right-hand side is only involving cosines, I'm going to just use the cosine identity, two cosine squared of x minus one. This is squared. We're going to subtract one. So we need to foil out what's in the brackets there. Again, so we're going to get a four cosine squared squared, which is the fourth power. Then we're going to get a negative two cosine squared. We're going to get another negative two cosine squared. And then we're going to get a plus one, like so, minus one. Let's see, combine some like terms. These cosine squareds combine together. So we're going to get negative four cosine squared of x plus one, minus one there. Then I distribute the two onto all of these pieces. So we're going to get an eight cosine to the fourth. Oops, I didn't write it down there correctly. That should be a four. So we get eight cosine to the fourth x minus eight cosine squared. And then we're going to get a plus two. But then we have a minus one for which when that minus one with the plus two interact, we end up with, of course, eight cosine to the fourth x minus eight cosine squared x. And then we're going to get a plus one, which you'll notice that is the right hand side, thus completing the trigonometric identity. And so we can use the double angle identity for cosine and all the settings where it might be appropriate, of course. But with cosine, you have some options. What's better? Do I want only cosines? Do I want only sines? Or do I want both cosine squared and sine squared? Depending on the context, you'll make that decision.