 Welcome back to our lecture series, Math 1060 Trigonometry for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misildine. In lecture 16, we're going to continue our discussion of trigonometric identities. Now that we've learned the fundamental identities and we've seen how we can use them to start proving trigonometric identities, we're going to start doing that in full force now. But before it can just jump into the deep into the pool, there's a few guidelines I want to mention that will help us as we prove them. These are not techniques for proving them, but these are guideposts that we should stay between so that we don't make mistakes while we are proving these trigonometric identities. The first one we actually did talk about before as one of our tips for proving trigonometric identities, but it's important enough that it's worth mentioning twice here. When you are trying to prove a trigonometric identity, work on one side of the equation at a time that is transfer the left-hand side into the right-hand side or transfer the right-hand side into the left-hand side. For more complicated identities, you can actually work on one side of the equation for a little bit and then move to the other side and then work on it for a while. Maybe it's find yourself in the middle, right? If you work on the left-hand side, it's like, I don't know how these are connected, then you work on the right-hand side, and then they can come together in the middle. Just you never, never, never, ever, let me say one more time, never use operations that affect both sides of the equation at the same time. That's a horrible mistake. You don't want to do that. Avoid that. We always work with one side at a time, but you can use both sides just never at the same time. The second guideline I want us to follow here is that it's usually best to work with the more complicated side first. We have these two sides of the trigonometric identity. I often choose the more complicated side to start working with that first, even if it's the right-hand side. To help you remember this, think about this. That in mathematics, unlike life, it is easier to simplify things than it is to complicate things. In life, we complicate things way too much, but it's easier to simplify than to complicate it. That is to take a simpler expression and expand it into a more complicated one. The third guideline I want us to follow is that when trying to prove trigonometric identities, look for trigonometric substitutions that involve the basic identities first. Look for things with the ratio, the reciprocal, the Pythagorean identities, maybe the symmetry identities. Use the basic identities first before you use the more exotic ones. Now, that's not going to be such a big deal right now in this lecture, but as we move on to lecture 17 and beyond, and we learn new identities, like the double angle identity, the half angle identity, the product of some identities, all of these, we should try to avoid using the more exotic identities until we really have no better option. That is, in medicine, when you're trying to give a prognosis or a diagnosis of a disease, you're told, don't look for zebras among the horses, right? If you see it, it's probably a horse, right? And so let's use the more basic identities first, right? When trying to prove trigonometric identities, it's also important to realize that algebra is your best friend. Look for algebraic operations that can help us, like addition, subtraction, multiplication, division. Could we add fractions by finding common denominator? Could we factor things? Could we cancel across a fraction bar? Could we distribute? Can we foil things? Turns out algebraic operations will be very useful, okay? Tip number five here, I should say guideline number five. If you cannot think of anything else to do, change everything to signs and cosigns and try to simplify it, because oftentimes proving that identity can be difficult, you have a cosecant here, you have a cotangent there, there's all these trigonometric functions with all of their trigonometric identities. If you can switch to signs and cosigns, then you have fewer identities you have to worry about and perhaps there's simplifications that can occur that were not obvious from before. And then the last guideline I'm gonna offer right now is always keep your eye on the side you're not working with to be sure you're working towards it or it's much simpler. Always keep your eye on the prize. When you're trying to do a trigonometric identity, you have the left-hand side, you have the right-hand side. Suppose you're trying to convert the left-hand side into the right-hand side. Well, you don't just start applying arbitrary identities to the left-hand side, you choose identities that will hopefully get you closer to the right-hand side. And so use your goal to help you guide in the direction you wanna go. And so I wanna prove a few trigonometric identities and show you how these guidelines help me as I try to prove it because I don't have these proofs memorized, I don't. I'm reproducing it in real time as we're recording this video. This isn't some scripted thing. How am I gonna do this? Well, when I look at this, I have to pick a side, left-hand side or right-hand side. Which one do I wanna prove? Well, the left-hand side is the product of two trigonometric functions. The right-hand side is just cosine. So the left-hand side looks more complicated. So that's the side I'm gonna start with. So let's take the left-hand side here. I actually always like to start by writing left-hand side and then I'll write whatever that left-hand side is. It helps me remember that I can only work with one side at a time and not doing two sides at the same time. That's a big no-no there, all right? So what can I do there? It's like, okay, shine and cosine. The right-hand side only has a cosine in it. Sorry, sine and cotangent. The right-hand side only has a cosine in it. So I have to somehow get rid of, well, basically what I'm trying to say is the right-hand side only involves sines and cosines, so to speak. The left-hand side does have this cotangent. So what if I were to replace the cotangent with cosine over sine, right? Because cotangent, excuse me, is cosine over sine. That's a ratio-identity error because somehow I have to introduce cosine into the problem, right? I have to somehow bring a cosine into it if I'm working with the left-hand side. How am I gonna get a cosine? Well, cotangent has a cosine in the numerator. Then when I look at that, like, oh, there's a sine in the numerator that cancels with the sine and denominator. I can use that algebraic simplification, all right? And then it just simplifies just to be cosine, which is the right-hand side. And I do like to write that out right-hand side there. And you put a little square there to indicate that the proof is done. You see this a lot of mathematics. They draw a little square at the end of the proof. It indicates that the proof is done. This is called the QED symbol. QED is an acronym of a Latin phrase that I will not, I won't bother saying it right now. But what it translates to English as is basically the question is now dead, meaning that the, I should say the doubt in the question is dead, right? Our doubt, whether this is a true statement or not, is now gone because we have the proof in front of us. And so to commemorate that our doubt is now dead, that little square is supposed to represent a tombstone. Yeah, it seems kind of morbid. Take that in though, but that's the etymology of that little square right there. It represents that the doubt is dead. And so we draw a tombstone, a little grave for our doubt. Isn't that so sweet? But how did we prove this thing, right? We chose the more complicated side, which is the left-hand side. We moved from the left-hand side one step at a time, always justifiable until we reached the right-hand side. We only did one side at a time. What are the justifications here? Well, the first one is just that is what the left-hand side is. The second one, we used the ratio identity, which again, our strategy was, we used only fundamental identities. We recognized that the right-hand side has a cosine. I have to get a cosine into the problem somehow. So I used the ratio identity, it did that. And then we did an algebraic simplification to get a cosine and then that is what the right-hand side is. So every step along the way was justifiable and each step was taken based upon the guidelines we learned on the previous slide. Let's do another example. Let's prove that secant theta minus cosine theta is equal to sine theta times tangent theta. All right, you look at that. I got some trigonometric functions. I got a difference of trig functions and I have a product of trig functions. I got to choose one side to get started with and then go from there. Now personally, I would think that the difference of trig functions is the more complicated side. The reason for that is that with products and fractions, it's easier to cancel terms. So the difference is actually a little bit more complicated for me, so I might start with that. So the left-hand side equals secant theta minus cosine theta, all right? What can I do with that? Well, the left-hand side is gonna be, it's sine and tangent. How am I gonna get a sine? How am I gonna get a tangent? You might not have any idea whatsoever. It's like, well, at the very least, I could write everything in terms of sines and cosines. The secant could become one over cosine, one over cosine theta minus cosine theta, like so. It's like, okay, but how am I gonna get rid of the difference? Well, if you're not sure what to do at this moment, it's like, well, I could find a common denominator, again, algebraic stuff here. I could find a common denominator. So I wanna cosine in the denominator here. I have to times the top by cosine as well. In which case then, this becomes a one minus cosine squared theta over cosine theta. In which case, maybe at this moment, you're stumped. You might not know what to do. It's like, I really don't see what's going on here. The other tip I would point out to you is that whenever there are squares involved, so notice we have a cosine squared now, it's very possible that a Pythagorean identity would be useful. So maybe I look up the Pythagorean identities. The mother identity tells us that cosine squared plus sine squared is equal to one. Well, notice that if I move the sine squared, excuse me, if I move the cosine squared to the side, this becomes one minus cosine squared theta is equal to sine squared theta. So that would get rid of the difference and would give me a sine. Would that be a good idea? Well, the left-hand side does have a sine in it. I need a sine squared, right? So that does seem like a step in the right direction, right? So let's try it. So we see that the left-hand side can become sine squared theta over cosine, like so. In which case, I got rid of the difference. I have just sines and cosines, but maybe you don't see where tangent is. Maybe you got stumped right there. Not a big deal. If ever you get stumped, start working on the right-hand side. It's like, okay, the right-hand side is equal to sine theta tangent theta, like so. In which case it's like, hmm, well, I have a sine and a tangent. Can I somehow make that look like sine squared over cosine? Well, if I switch maybe tangent into sines and cosines, so I'll just give her a tangent. Just switch to sines and cosines. You're gonna end up with sine theta times sine theta over cosine by the ratio identity. And then I look at that, it's like, oh, sine times sine is sine squared. Oh, oh, oh, those are equal. Those are equal. I found them in the middle. Yay, we get excited. So that's the thing is if you get stuck in the middle, you can sometimes work on the other side and connect them in the middle somewhere. So notice I never worked with both sides of the equations at the same time. I worked from the left-hand side until I got here. I worked from the right-hand side until they met together, right? And so that then proves our identity. Let's put a little tombstone because our doubt is now dead. Let's do one more example of this. All right, let's take tangent. Let's prove the identity of the tangent theta plus cosine, excuse me, tangent of x plus cosine of x is equal to sine x times secant x plus cotangent x. The name of the angle really doesn't matter too much. Call it x, call it theta, whatever. It is important you do write it. Don't write things like tangent plus cosine is equal to sine times secant plus cotangent. Don't do things like this. The variable does actually matter. I mean, its label doesn't matter, but it is part of the function. We do need to include it. It's bad notation to not write the angles on it. If I was grading an exam and people didn't write the angles, I would dock them some points because that's not proper notation there. So just to watch out for such a thing. All right, so let's prove this identity. What can we do? Well, we have a sum of two trig functions on the left and a product of two trig functions, which one of those functions is itself a plus. I said that addition was more complicated than multiplication, but the left-hand side has addition, but the right-hand side has addition and multiplication. So I definitely would say that the right-hand side is the more complicated one here. So I'm gonna actually start with the right-hand side. Okay? The right-hand side also has a very natural algebraic operation that seems like it'd be useful. What if we distribute the sine, right? Because after all, the left-hand side is just a plus. We have a plus in a product. You have to get rid of the product. Distribution could get rid of the multiplication, so to speak. That would then give us sine x times secant x, and then you're gonna get a sine x times cotangent of x. So if you're not sure how to get this to look like tangent and cosine, well, maybe we just switch things to sines and cosines, right? So the secant is gonna become a one over cosine, all right? And then we have a sine. Well, let's just keep that around, but then the cotangent will become a cosine over sine. See what we can do with that. Well, I see that with the second product, there's a sine over sine, so let's cancel out, and that would then give me a cosine, right? It's like, oh, looking at what I'm trying to get, there's a cosine there. Do I have a tangent over here? Well, if you look at the first part, you have a sine over cosine. Oh, oh, oh, sine over cosine, that's a tangent, right? And so then this thin is in fact, tangent x plus cosine x, which is equal to the left-hand side. So kill the doubt, bury it in the ground, and we've now proven this trigonometric identity. So hopefully these examples illustrate how following the guidelines we started off this video with truly do help us prove these trigonometric identities. Watch some of the subsequent videos for this lecture as well, so you can see how we can use some more complicated trigonometric identities, but still following those guidelines, we will be successful in proving these trigonometric identities. If you follow these guidelines and with sufficient practice, you'll be able to prove these things too. I know you can.