 It says this, in mathematics, it is important to understand the difference between an equation and an identity. 2x squared plus 3 equals 11 is an equation, and that's because it's only true for certain values, negative 2 and 2 work nothing else does. x plus 1 all squared equals x squared plus 2x plus 1 is an identity. It's true for all legal values of the variable x, and there are all sorts of identities out there. I'll give you one for example that we learned, we learned that the log base a of b plus the log base a of c is the same as the log base a of, we said, oh, adding two logs is the same as multiplying. That's an identity. Now it's only true for legal values, which means that b has to be positive and c has to be positive, but it is true for any positive values of b and c. All sorts of identities out there. What we're going to do is we're going to derive a couple of basic trig ones, and then we're going to use those to prove a lot more. And when we're doing a proof, there's a very simple rule when you're trying to write out a proof. If you think it, write it. If you think it, write it. I've marked these on provincials, and there's always kids who do too much in their head, and unfortunately we have to take marks off because you didn't tell us how you got this step. You're trying to learn, Tyler, how to do an ironclad argument. So I would actually argue as a skill, this is fairly useful, and if you're thinking about going into law, this is a lot of what a good lawyer tries to do anyways. We have the basic trig identities, y over r, x over r, y over x, and x squared plus y squared equals r squared. We're going to use these for a few minutes, but then we're going to chuck them and never use them again, not for doing identities. We're going to use them to develop some more useful ones. We also have the reciprocal ones, and I think you will notice these are on your formula sheet. Are they not, I think, second or third row? Are they not, I think, second or third row, yes? Yes, they are, yes? Okay, you found them? Making sure you know they're there. I made you memorize them because I've learned over the years kids that have to refer to the sheet for those have not done the homework and are going to flunk. But they're there. We can use the basic and the reciprocal trig identities to prove the quotient, and in particular the Pythagorean identities. These two are the ones we're going to be using like crazy. It says, when verifying an identity, turn the page, we have to treat the left side and we'll often abbreviate that as L, S. And the right side, we'll often abbreviate that as RS, separately until both sides represent the same value. Even though there isn't, if you look at example B, even though there is an equal sign here, I can't do my normal equation solving rules. I cannot, for example, minus one from both sides, even if that might be hugely convenient because I'd be assuming they were equal in order to prove that they were equal. But none of my equation solving rules are usable anymore. I have to work on one side or the other side and try and make the one side look just like the other side. So these are sort of brain teasers. These are sort of puzzles. Example one says verify the following identities for the value given. I'm just going to tell you for what it's worth, if you put a 60 degrees there and there and there, it will work. These ones, we're not really going to be doing. I want to get to proving identities. We can use identities to derive other identities. And when proving an identity, we must treat the left side and the right side separately. Ian, what we're going to do very often is what's called a T table. We're going to draw a great big capital letter T. And this line down the middle is telling me, unlike an equation, I can't move back and forth from side to side. It would be nice and there's going to be somewhere you're going, you know, all I want to do is add this over to the other side and I'm done. You can't. And what this one wants us to do, it wants us to prove that tangent is the same as sign over cosine. Oh, by the way, it adds a restriction that says cosine can't be zero. Why can't cosine be zero? Okay, so remember I said for that log one that I made up earlier, the restrictions where my logs can't take a log of a negative, my values had to be positive. There are often for identities restrictions for this one, cos can't be zero because you can't divide by zero. So they've already added that restriction. Okay. Generally, I start with the uglier side first. What looks uglier, that side or that side? The right side, I think it's got a fraction. Okay, let's start with that. And just this once, Dylan, we're going to go back to x, y and r. So sine is what over what? Y over r over cosine is what over what, Miguel or that Tyler? X over r. How do I divide by fraction? So this is the same as y over x, sorry, y over x, Mr. DeWitt. Y over r times r over x. And how do you multiply fractions? Top times, top, bottom times. How many r's do I have on top? One, how many r's do I have on the bottom? And I end up with y over x and oh, pray tell, what is y over x? So I would put an equal sign there, an equal sign there, and my final step, I would say this is tan x. There is a tradition hundreds of years old when you have done a proof, if you're a nerd, to write q, period, e, period, d, period. You may have heard the term q, e, d. It's Latin. I think if my Latin is close, it means which was to be demonstrated. And so back in the Middle Ages, when they were doing the math, they would often write that at the end of a proof. In university, we used to joke that it stood for question easily done and was the biggest lie on earth. In university, I was doing proofs that were three and four pages, not three and four lines. It's an art, you get to it. You're going to find Alex, I will teach you the science of identities. And most of you will develop the art of it. It's in our brains. In fact, psychologists have argued that problem-solving has replaced hunting in our hunter-gatherer evolutionary background. That we get the same adrenaline rush now when we see the solution to a problem as our ancestors 100,000 years ago got when they threw the spear. Kind of creepy, kind of cool. We have just derived a new identity. And this is the one that we're going to memorize. This. It is on your formula sheet. Is it not? It does say that tangent is sign over cosine. Is that correct? I'm going to tell you right now, the kids that don't memorize that are flunking the test. Now, you're not going to have to go out of your way to memorize it. This is going to be one we're going to use so often you will. But I'm going to warn you if test time comes. Oh, by the way, if tangent is sign over cosine, what's cotangent? Cos over side. So you just added two identities and you can now use those. You will never have to fall back to x, y, and r. We never will. We'll use sine and cosine and tangent. Turn the page or next page over. It says, use the basic identities to prove that 1 plus 10 squared equals ck times. I'm not going to do that. We can use x and y and r to do it quite easily. But my concern is you're going to think, oh, that's how we're going to do identities. We don't use x, y, and r. So I'm going to highlight that, though, because I believe that's on the top of your formula sheet. Is it not? Third one over or second one over on the top? Yes. By the way, keep a formula sheet in front of you for this lesson. Little hint. Is that yes? Yes. 1 plus 10 squared equals ck times squared. I don't memorize that. I don't have that one memorized. It's on my sheet. Oh, I do know if I ever see a secant squared. Hey, that showed up somewhere. Or if I ever see a tan squared. Hey, that showed up somewhere. I know that, but I don't know where it should. I go find it. So these are the basic identities. We have the quotient identities. They're on your formula sheet. And then we have the pythagorean identities, the big one. And I'm going to change colors for it. Is this one. Sine squared plus cos squared equals one. It's a fairly easy proof to do. We may end up doing it in the homework, I think. That's one that you will memorize. Because you'll use it over and over and over. But please understand, this actually contains three equations. If sine squared plus cos squared equals one, get the cos squared by itself. What's cos squared the same as? To the math. Yeah. One minus sine squared. What's sine squared the same as? One minus cos squared. These other two are also called pythagorean. By the way, can you see why they're called pythagorean? There's something squared plus something squared equals something squared. The one doesn't have a squared on it. One squared is one. Oh yeah. So these are the pythagorean ones. I don't have those two memorized. I got to be honest. If I see a cosy can't squared somewhere in a question, Jesse. You know, that's on my sheet somewhere. I'll go hunting for it. Oh, and by the way, what is cotangent squared the same as? Get it by itself. Cosy can't squared minus one. What is tan squared the same as? C can't squared minus one. So each of these, Jesse, contains three other equations. This one I memorized and it's fairly easy. Sine squared plus cos squared equals one. Okay? So these identities are on the formula sheet. We can use these guys and these guys to prove a bunch more. And now we're going to get into the water. Up until now, I've been treading water and you guys have been on the edge of the pool. Are you ready? Take a deep mental breath. Jump in. The water is going to be great. It's going to feel cold at first, but area guarantee within a day or two, is what Michael Phelps would be. Okay? So it says this. We use the basic identities in terms of x, y and r only to prove the quotient and pythagorean. We're not going to use them again. Turn the page. So here is some of the questions that you'll get. And again, you want that formula sheet in front of you. It says this. Express each as a single trig ratio. Then it says use a graphing calculator to verify. I'm not going to do that. We're going to do the algebra. Now this says sine squared over cos squared plus one. Apparently I can write that as a single trig ratio. Well, what was sine over cosine? Have you memorized it already? What was sine over cosine? So what do you think sine squared over cos squared is? Okay. This is equal to tan squared x plus one. And I haven't memorized this, but I know when I see tan squared, that somewhere on my sheet. Go hunt for it. What's tan squared plus one the same as? Oh, have we represented this expression as a single trig ratio? Never done. This would be the kind of multiple choice question you might get. Write this as a single trig ratio. Okay. Rules. If I'm doing a T table, I start with the most complex or the ugliest side first. This is not a T table. They haven't given me two sides. Second rule. Okay. The white piece of paper that I gave you. What was the second hint that I gave you? Stop. Read it louder. Please, Pat. Stop. One, two, three. Plus, what's cotangent in terms of sine and cosine? Cos over sine? Did I just write a fraction? Then I want the next term to be a fraction. Cos x over one. Oh. How do I multiply fractions? Top times, bottom times. Okay, let's do that. So I have sine x over one. If I have one fraction, I want everything to be fractions. Plus, cos squared x, because cos times cos is cos squared. Yes. Over sine x. How do I add fractions together? I need a common denominator. What's on the bottom of my first fraction? One. What's on the bottom of my second fraction? Okay. I'm joking with you all year about how you suck at fractions, and then you'll notice after Christmas I moved to mediocre at fractions. This is where you're going to get tolerable at fractions. This is where you won't make me gag at fractions. Some of you may even get proficient. I don't know if you'll get good. Some of you may get proficient. We need to find a common denominator. What's my common denominator going to be here? Sine. Multiply a one by to change it into a sine. Same to the top and bottom. I'm going to multiply the top by sine x and the bottom by sine x, and you must write that. Stop. Look up. Some of you were taught wrong or lazily in grade eight or elementary school, and you're just going to do that. I'm taking marks off because that is wrong. You're not multiplying by sine. You are multiplying by... One. Sine over sine. One. You're not changing the question, and if there's one common error I need to beat out of you, it's the temptation to multiply by something other than a one. All we're going to be doing is multiplying by ones. Funny looking ones, but ones. So I'm going to be really, really fussy, and I'm going to say, make sure you write that in. Have you seen it yet? Did you get your nerdy adrenaline rush yet? You see it? Sine squared. And then this is already over sine x? Yes? So it's just going to end up there. Have you seen it yet? Have you seen it? You see it yet? What is sine squared plus cos squared? This is one over sine x. And I'll pray tell, what is one over sine x as a single solitary trig function? Cosy kin. There you go. You're going to find when you get good at this, three lines, or three lines, or if you get really good four lines from the end, suddenly you'll see it. I hope you get that nerdy adrenaline rush. I hope, I hope, I hope. Example two. Express two tan a, all over one plus tan squared in terms of sine a and cos a, and write in simplest form. All right. How many trig functions do I have in this question? Two? Tell me what two they are, please. Ten. Sorry. Ready? Erwin, what trig function do I have in this question? Say it with a big loud voice for once, my friend. Ten. Darn right. So Pat, how many trig functions do I have in this question? One. So I'm not going to write everything in terms of sine and cos. And I'm, you know what? The bottom there is screaming out at me. One plus tan squared. That's familiar. What? Two. Tan a, all over secant squared a. How's that help? How many trig functions do I have now? Two. This is the kind of halfway area. When I have two, if I can't do anything else, I'll rewrite everything in terms of sine and cos. So you're ready? What's tangent in terms of sine and cos? Two. What's two tangent in terms of sine and cos? Two sine over cos. Not two over two. Not two over two. The two goes on top. All over. What's secant in terms of sine and cos? One over cos. What's secant squared in terms of sine and cos? One over cos squared. Okay. Wait a minute. Isn't this one fraction divided by another fraction? Is that why you snuck this in last unit? How do I divide by a fraction? It's going to be two sine a over cos a times cos squared a over one. Right? How many cos do I have on top? Two. How many on the bottom? How many left and where? And in fact, this simplifies to two sine a cos a. Now, did we match the instructions? It wanted us to write it in terms of sine and cos. Have we written it in terms of sine and cos? Have we written it in simplest form? Do you see any squares or anything that we could do magic with? Now, you don't know this yet, but see if you can find it on your formula sheet and you can find it on your formula sheet. So, this is actually the sine of two theta. This is actually a period horizontal compression by a factor of we're not going to do double angles yet. We'll get to those. But for what it's worth, eventually when you see that, you're like, I know that somewhere, we're just going to be dealing with the top two or three rows, the Pythagorean and the reciprocals for really quite a while before we bring in the sum and difference and the double angle identities. It says factor. It says complete the next questions. What we are going to actually do is turn please to the next lesson. Lesson six. What I'm going to be doing for the next couple of days is actually it's a lot of talking. I'm going to be working through a few of these with you each day and you're going to find all of a sudden different ones of you are going to be going, I got this. What I've done, actually what another teacher did is they've gone through every provincial exam going back to 1994 and they lifted the trig identities electronically. I have a 12 page, 25 trig identity big package for you. If you get through that and all of you will, it's going to be one of the homework assignments for this unit, you're in really good shape. So you're ready? Oh, it gives you hints. My hints are better. I got an extra one. It says, consider this statement here. First of all, it says verify the statement is true. It is, I'm not going to do that. But if you put a pi by three on your calculator there and there, type this in and press equals. You get the same decimal as if you put a pi by three there and a pi by three there, type that in and press equals. B, it says use a graphing calculator. I'm not going to do that. But for what it's worth, Jesse, if you graph this as Y1 as Y2, only one graph would show up on your screen. It'd be the same line overlapping twice. What we are going to do is prove these algebraically. So I need to copy this right here. You guys can too. We're going to write one over cos theta minus cos theta. And we're trying to show that it's equal to sin theta tan theta. Here is our first identity. You got that little hint sheet. By the way, you want the hint sheet and the trig identities formula sheet in front of you somewhere on your desk. So the hints, I said this, start with the more complex side first. This is a bit of an unusual one, one Victoria. Both of these sides look ugly to me. This one a bit more so because of fractions, but they both look ugly. What that really tells me is I'm probably going to have to work on both sides along the way. What did step two say, Victoria? What I'm really be looking for is like squares. That was step two. Okay, step, what, I thought that seemed wrong. What a step, not from the workbook, from my hint sheet, which I said is far better than the workbook. Ready, scene one, act one, take two. What does step two say? Stop, I see three. I think the first thing I'm going to do is rewrite this in terms of sine and cos. Is this already in terms of cos? Leave it. Is this already in terms of cos? Leave it, ooh. Sine theta, what is tangent? Sine theta over cos theta. Did I just get a fraction? Then I want everything to be fractions. Never have some and some not. In fact, I'll be honest, look up, look up. This is driving me crazy. I have to do that. That is just making things so much tougher that I don't clearly see what's on top and what's on the bottom. By the way, what's another way to write sine times sine? This side here works out to sine squared theta over cos, and I think I'm stuck. I'm going to stop there. When you're doing identities, one of the best things you can do is always move your eyes from the left side to the right side, glance at the left, glance at the right, because I've got to see how many fractions do I have on the right side? One, how many fractions do I have on the left side? You know what I should probably do? Try writing this as one. Do you see how I figured that out just by looking? So, we're adding and subtracting fractions. We need a common denominator. What's my common denominator going to be? Stop. Look up. Here, I'm not done, but I now know I'm on the right track. I have to be. That can't be a fluke. In fact, I did a bunch of this already in my head. I got my nerdy adrenaline rush about as soon as I wrote that down. Maybe you see it yet? I don't know. I want this to be over cosine. Is it already over cosine? And we just drop it down. Oh, and the minus sign would drop and change it into a cosine. Top and bottom. I've yelled at you about just writing it on the top. I'll change colors. I'm going to put a cos there and a cos there. So that puts this over cos. And what is cos cos? Do you see it yet? Do you see it yet? Do you see it yet? What is one minus cos squared? Look at your sheet. What? Director, you guys, I had a student from Germany last year who was doing Latin. I don't know if you guys are. Am I close? Pardon me? Okay. Am I close? It's quarter. But I've never heard it pronounced in English. So I don't know quite how it goes. Oh, the Americans will put a little by the way, they'll color in a square because the Americans don't like having to learn a second to me. It's me screaming at the world. Success. Turn the page. Oh, sorry. Don't turn the page. Go back. Restrictions. Cos can't be zero. Right? No restrictions. No restriction. Tangent always has the restriction in built in because tangent is sign over cos sign. Cos sign can't be zero. Oh, I already said that. Okay. Now turn the page. Okay. It says consider this one. Check it for a variable. It's true. I'm not going to bother doing that. Use a graphing calculator. Trust me. If you graph this and graph that, they're the same graph. Forget it. I want to do this algebraically. So I have sin theta over tan theta. I want to show that that equals cos theta. Let's begin. I think I'll be tackling this side. In fact, I don't think I can do anything with the right hand side. I think cos theta is just going to stay cos theta. So when it says begin with the uglier side first here. Victoria, what was hint number two from my sheet, please? Stop. One, two, three. Oh, yes. Keep reading. Rewrite. Rewrite everything in terms of sin and cos. Okay. This is going to be sin theta all over what's tangent in terms of sin and cos. Oh, how do I divide by that fraction? So this is going to be sin theta times cos theta over sin theta. Oh, I hate the way this looks. If I have one fraction, I want everything to be a fraction. So I can clearly see what's on the top and what's on the bottom. And how do I multiply fractions? Oh, pray tell. Top times, bottom times. How many signs do I have on the top? How many signs do I have on the bottom? Dylan, technically over one, do I need to write the over one? Say no. I have cos theta equals cos theta. Got to be honest, those first two are barely in the kiddie pool. Shallow end though. Now, here we go. Oh, restrictions. tan theta can't be zero because I can't divide by zero. And tangent has its built-in restriction as well because tangent has a denominator that's invisible. What's always in the denominator of tangent? cos theta can't be zero. Sometimes on the multiple choice, all we'll ask for are the restrictions because they figure if you can find those, you can break the identity apart. So that's why I'm practicing those. Example three. Ha! Here we go. Let's prove this. Suggestions. What was hint number one? I think I'm doing most of my work on the left side. Yes? Okay. What was hint number two? I have more than two trig functions. Although the left side is all sine and cos, what's one divided by cosy-cant? If sine is, sorry, if cosy-cant is one over sine, one over cosy-cant, or cosy-cant, flipping cosy-cant, we'll flip the one over sine. This is just sine x. And I'm pretty sure I'm done on the right side. Now Matt is scouring his sheet, wildly looking for stuff. Matt, you got the white sheet in front of you? The hint sheet, the hint sheet, the hint sheet? What's hint number three? Do we have any identities that have a cubed in them? So I'm definitely not using this. We do have some with a squared, but not a sine without a squared and a cos with a squared. What's step four, Matt? What? What's the first thing that you always, always, always, always, always, always? Is this why you worked that in, Mr. Dewick? Darn right. Always look for. There is one. You see it? You see it? You see it? You see it? What? And when I factor a sine x out of the first term, Matt, what's going to be left behind? Stop, stop. When I factor a sine out of the first term, what's going to be left behind? Cos squared, and the plus sign would drop down. And when I factor a sine out of a sine cubed, by the way, have you got it? You see it? Have you got it? What's going to be left behind? And Matt, oh, pray tell, what is cos squared plus sine squared? Now you would have to show that step. Shallow end ain't so bad. Water's kind of nice. Turn the page. You ever been in a pool? Start in the shallow end. You start walking, you start walking, you start walking, and all of a sudden you come in your child's life to a very significant moment. What? The drop off. Right? The pool, all of a sudden, goes steep. Here it is. It says, prove that this is an identity. That's my short abbreviated way to do a t-table, by the way. I think it's fairly obvious, which is the uglier side. We're going to be working most of it on the left. How many different trig functions? How many different trig functions? We're going to, first thing, rewrite everything in terms of sine and cos. But since I'm going to be spending most of my time on the left, and since I noticed there's only one thing here, I'm going to tackle this first. What if I just said cotangent? What's cotangent in terms of sine and cos? Cos over sine. What's one over cotangent? What's on top on the right-hand side here? What's on the bottom? As I'm working this, if suddenly a cos sign shows up on the bottom, I'll probably smile. Or if suddenly a sign shows up on the top, I'll probably smile. Sine a, plus, what am I going to write instead of tangent? Did I just write a fraction? Stop everything a fraction. I will never have some and some not. All over. Stop everything a fraction. I will never have some and some not. Now that looks ugly, but I want to show you a great trick. It's called clearing complex fractions. I'm going to zoom in here. This is a complex fraction. What's a complex fraction? A fraction on top of a fraction. Now, technically, this was also a complex fraction. Technically, where was it, Mr. Dewick? It was on the previous lesson. I'm not going to go finding it. Technically, because this is sine over one, this is a complex fraction. What I really mean by complex fraction is, not when you have a single solitary fraction over a single solitary fraction, how do I divide by fraction with multiple? But when you have multiple fractions over multiple fractions, there is a way in one line to get rid of all of the yuckiness. You may have been shown this skill in grade 10 or not. What you do is you look at each small fraction, this guy, this guy, this guy, and this guy, and you look at the denominator. Call it out. Call it out. Call it out. Call it out. You're going to multiply the top and the bottom by that common denominator. What is the common denominator of all those minis? In this case, cos a. If I had a cos squared here, it would be cos squared. Or if I had a sine here, it would be sine a cos a. But I'm going to multiply, again, the top and the bottom. Don't write this down. Watch. Alex, put your pen down. I'm trying to avoid a bunch of common mistakes. Now, when I say multiply the top and bottom, a lot of students do this. Okay. Don't write this down. This is dumb. And they've just confused themselves because how many levels does this fraction have? Four. See it? How many levels did they write here? Dumb. I'm going to multiply the top and bottom by the common denominator. But I'm going to write it like this. Cos a over one over cos a over one. So I can clearly see what's on the top on the top and what's on the bottom on the top. So I can clearly see what's on the top in the bottom row and what's on the bottom in the bottom row. Now, is that still cos over cos? Yes. Is that still technically a one? Yes. So am I changing anything? No. Don't write this down yet. I know Alex, you want to reach for your pencil. Relax. If you watch it and then you rewrite it because I'm going to freeze the screen, you can try and do it from memory and we're going to reinforce. Don't write this down yet. But here's what happens. I multiply. What's cos a times sin a over one times one? Well, that's going to work out to sin a cos a. I'm okay with that. Plus what's cos a times sin a over one times cos a? Or how many cos is on top? How many cos is on the bottom? Sin a. All over. So I've done this. I've done that. I'm going to do this. I'm going to do that. What's cos a times one? Plus what's cos a times cos? Now that may look uglier to you. It's not. I guarantee it. First of all, how many levels do we have now? Nicer. Does anybody see it yet, by the way? GCF on the top, GCF on the bottom. What's your GCF on the top? What's your GCF on the bottom? I wonder if the rest of the cancer we'll find out. Anyways, now write that down. That's how you get rid of a complex fraction. You find what your common denominator is of all the little mini denominators and you multiply the top and bottom by that. But please, trust me, put it over one and over one so you're multiplying by four levels. You'll make far less dumb mistakes along the way. And once you've written that down, put your pencils down again and look up so that I know you've got it copied out. But I'll wait till everyone has it copied out. Good? Good? More waiting? No problem. There is another way to do this. What you could have done is combine this as one fraction by finding a common denominator. Combine this as one fraction by finding a common denominator. And then you would have had one ugly fraction over one ugly fraction. How do I divide by fraction, flip it and multiply? It will get you to this same line. But what I did in one will take you three lines. And I just worry that the more lines, the more room there is for sloppy mistakes. And that's, I think we've noticed in Math 12, our biggest enemy on tests is sloppy mistakes, right? So this complex fraction trick is well worth learning. Now pick up your pens again, GCF. And I'll have a cos A plus sin A sin over GCF. Oh, just be one. Thank you, it would be. Did I say that the biggest enemy is sloppy mistakes? Yes. And I'll factor out a cos A in the bottom. And I get one plus cos. Can I cancel those? Are those identical brackets? Is one plus cos the same as cos plus one? Yes. Now if it was one minus cos, that's not the same as cos minus one. Order makes a difference when you're subtracting, but not when you're adding. This cancels, this cancels. And hopefully you'd clue in that it had to because what do I have sitting by itself right here? What do I have sitting by itself right here? One more. And then I'm going to give you the big massive assignment and I'll tell you a few you can start to work on. But it's going to take a little bit of practice. I realize that. Five. Prove this. Okay. Which side's uglier? Left side. Although that right side, I think, well, I will say this. Cosy can't squared. Is that on your formula sheet somewhere at the top? What? Okay, I'm just going to kind of keep in mind that that's one plus cotin squared. If I end up with that over here, we'll see. How many trig functions in this equation? In this identity? I only see two. So although I was initially thinking about writing cosy can't squared as one over sin squared, I'm not going to just yet. I see a squared and I see a one. That's screaming at me because I have a bunch of squareds and one along the top of my formula sheet. What is c can't squared minus one the same as? Oh, I like that. So you're telling me this is c can't squared theta all over tan squared theta. Okay. Oh. Now how many trig functions do I have in this equation? Now let's try rewriting everything in terms of sin and cos. So c can't goes with cos c can't squared will be one over cos squared all over. Tan squared, well tangent is sin over cos. So tan squared is going to be sin squared over cos squared. And cosy can't is one over sin squared. Suggestions? Flip and multiply. Yeah, okay, let's try that. Have you seen it yet? Have you got nerdy adrenaline rush? You see it? You see it? You see it? We're going to get this. One over cos squared theta times cos squared theta over sin squared theta. And how many cos squareds do I have on the top? One. Well, oh, sorry. How many cos do I have on the top? Two. How many cos do I have on the bottom? Two. How many cos squareds do I have on the top? One. How many cos squareds on the bottom? One. Regardless, woohoo. And lo and behold, are these the same? There's a science and an art to these. You're going to see some, oh, rule of thumb, by the way, usually the uglier they look, the easier they fall apart. Some of the toughest identities look simple to start out, but because they look so simple, there's nothing to work with. Usually if it's complicated, there's like eight different ways to get to the final answer and you're going to find one of them. Okay? It says, using identities to solve equations, we're not going to do that. Instead, I have a lovely present for you.