 In lecture 15, we began our official study of trigonometric identities. We reviewed the fundamental identities we've seen previously in the lecture series, and we've used them to simplify both trigonometric and algebraic expressions. So what our goal is, and one of the big goals of chapter six here, is for us to learn how to prove our own trigonometric identities. And so this often is intimidating for many trigonometry students. And I mean, it's unfortunate thing because trigonometric identities are our friends, they're our tools. Basically, what we're saying is every trigonometric identity is a tool we can add to our toolbox. So do we want a toolbox that only has like a hammer and a screwdriver? Or do we want a hammer, a screwdriver, a ratchet, some wrenches, a saw? I don't know. I mean, there's so many tools. You go to a store like Home Depot or Lowe's and just go down the hardware area. You can see all these tools. It's like, wow, I need a square. That's so awesome. Oh, I need that crowbar. I often do home renovation projects in my home. And so I love and excuses like, ooh, I gotta go buy a reciprocating saw because I have a job that requires it. So every time I do a new job, I basically get a new tool, which makes my toolbox bigger and bigger, bigger, bigger so that I can do better and better, better jobs. My jobs become easier the more tools I have. And that's how trigonometric identities work as well. Now, unfortunately, if you have a very obscure tool, like what in the world is an Allen wrench, right? What's a hex key? They're the same thing. But, you know, but if you don't know how to use it, then it's not a very useful tool. We need to learn how to use them. And so in this case, we need to learn how to prove these trigonometric identities. And so in that regard, what I'm gonna do for the remainder of this lecture 15 is give us pro tips on proving trigonometric identities. We'll do that in force in the next lecture, of course. And so each of these videos will be one tip at a time so we can drill in these ideas of how to prove trigonometric identities. So the first tip, which you can see here on the screen, when proving a trigonometric identity, always work one side of the equation at a time. That is transform the left-hand side into the right-hand side or transform the right-hand side into the left-hand side. We never, ever, ever, ever, ever, ever one more time, never use operations that involve both sides of the equations at the same time. So before we prove the trigonometric identity, let me show you why we never do operations on both sides of the equation at the same time. Let's prove the following. I have a theorem for you. The theorem is that one equals negative one. Now you might be laughing at me right now. This is a false statement, right? One's not equal to negative one. Well, here's the proof. The proof is one equals negative one. I'm then gonna square both sides in which case one squared is equal to one. And then negative one squared is also equal to negative one. Oh, well, negative one, or one equals one is definitely a true statement. So since this statement is true, that must imply that the original statement was true. And that's the proof. That's a false proof, right? That is not true. Don't do that, right? But that illustrates a point. This is how many students try to prove trigonometric identities. And the thing is, if this is a valid proof technique, then I can prove a false statement that then becomes true. Well, that's not the case. It means the technique is invalid. So any truth we discover using an invalid technique actually doesn't verify the truth of it at all. So we don't work simultaneously with both sides of the equations. So what someone would do, conversely, like with a situation like this, is like, oh, cosine tangent equals, cosine tangent equals sine. Well, let's see what we could do there. We could divide both sides by cosine theta, right? The cosine there canceled. Then you get tangent theta equals sine theta divided by cosine theta. Oh, tangent equals sine over cosine. That's the ratio identity. Boom, I'm done. So they would say that's the trigonometric proof, which that's not a valid proof. That's the same problem about proving one is equal to negative one. Basically, the issue here is it's circular reasoning. If you start using this equation, like if you start working on both sides of the equation simultaneously, you're assuming it's an equation, because those techniques are only valid to equations. In which case, if you assume it's an equation, you assume it's true, right? In which case, then you're assuming it's true, then to get to a statement that's also true, and then you conclude the truthfulness of the original statement. This is circular reasoning. It's not a valid logical argument. So instead, what we have to do is we have to work with, pick one side. In this case, we're gonna pick the left-hand side. So pick the left-hand side, cosine. Oh, so that's just cosine theta, tangent theta. And we have to then show that the left-hand side is equal to the right-hand side by a sequence of identities. So we might say something like, well, what if I change tangent into sine over cosine? So we get cosine times sine theta equals cosine, or sorry, sine theta over cosine theta. So these equations can be justified. So this first equation is justified by the ratio identity. All right, then the next one is just, well, cosine cancels out the cosine, and we get that this is equal to sine theta, like so. And what's the justification here? Well, this is just an algebraic simplification. You just simplify the expression by canceling out the cosines. And then you'll notice that sine theta is equal to the right-hand side. And so we've showed that the left-hand side is equal to the right-hand side as a sequence of equalities. And you can write those equalities horizontally, or you can write them vertically, or a combination of the two, doesn't matter. But the point is you move from the left-hand side to the right-hand side. You don't work with both sides simultaneously. You just take one step at a time, one step, two step, red step, blue step, things like that. You just do one step at a time. The ratio identity was applied. An algebraic simplification was applied. And that showed that the two quantities were in fact equal to each other. It's important you only work with one side at a time and move from that side to the other side. So that's trigonometric identity tip number one.