Identities Trigonometry DIFFICULT examples Grade 11

追加: 2026-05-15

[00:00:00]In this lesson we're going to look at more difficult or advanced identities.

[00:00:05]In order to prove these two identities, we need to do something different to what we did in the previous lessons.

[00:00:11]Something a little bit weird that maybe you wouldn't necessarily normally do. We do this when nothing else works. Using basic identities doesn't work. Sometimes simplifying, factorizing, none of it's working. So I'm going to show you what you do if you see ones like this in your exam.

[00:00:28]Of course the question starts by saying prove that. Remember if you want to pause the screen and try it first, please do so.

[00:00:35]Okay, so I'm going to start with the left-hand side because there's more to be done on the left-hand side. It looks a little bit juicier to work with. So I'm going to say left-hand side and then as you know if you've watched my identities videos in the past, I like to just rewrite the left-hand side quickly.

[00:00:51]So immediately what I would think about doing first is difference of two squares here in the numerator because we've got two terms. There's a minus in between and these are both square um even numbers. So what we could do is we could say cos theta plus sin theta, cos theta minus sin theta.

[00:01:17]And already you can see that I've got something similar on the right-hand side in my denominator over here which means I'm probably on the right track.

[00:01:25]Remember if you get thrown off oopsie.

[00:01:28]If you get thrown off but how to deal with something like this, imagine if it were algebra. If we had something like x squared minus y squared, you would know to factorize that. You would say oh it's difference of two squares. x plus y, x minus y. So it works exactly the same with trigonometry. The problem comes in with this denominator over here. This this one over here. Think about it. What can I do over here?

[00:01:52]You can't take out a highest common factor.

[00:01:55]You can't do difference of two squares because two is not a perfect square.

[00:02:01]Okay, so what on earth are we going to do? It's not a trinomial, it's only got two terms. So, in a case like this, the only way we're going we're going to be able to move forward is to replace the one over here with remember one our basic identity one is equal to sin squared theta plus cos squared theta. This was one of the two basic identities that we learned in grade 11.

[00:02:27]So, what we're going to do is instead of one, I'm going to write sin squared theta plus cos squared theta and then this minus two sin theta cos theta stays. That doesn't go away and I don't know if you can see or envision what we're going to do at the bottom now. But one thing that I do want you to notice and maybe you'll figure it out then is we've gone from two terms at the bottom where we couldn't do HCF, trinomial, difference of two squares, we've gone to three terms. And where does your brain go in terms of factorizing when you see three terms? I hope that you're saying to me trinomial. But how would you do a trinomial here? Maybe it helps for you to see it like this with the sin squared or let's do the cos squared theta first. Cos squared theta, so I'm just taking this term and I'm putting it first and then I'm going to make this my middle term minus two sin theta cos theta and then I'm making this my last term plus cuz there's an invisible plus sign here, plus sin squared theta. Again, if this is throwing you off, would you be able to factorize that bottom? Look at the Look at the denominator. Remember, oh, there's a lot a lot going on here, but this is a fraction that I'm looking at here.

[00:03:46]The denominator. Would you feel comfortable with factorizing something like that if it who like this? x squared minus 2 xy plus y squared. You need to be able to factorize something like that, by the way. That would be x minus y, and it would be x minus y. Because if you have to multiply that out, you would get this over here. So, how would we then apply that to the trigonometry of it all?

[00:04:12]So, you keep the top the same, and then we're going to open two brackets because we are basically doing a trinomial. You put cos theta cos theta. Because again, remember, if you had to multiply this out, cos theta times cos theta is cos squared theta.

[00:04:29]And then we're going to Oopsie, don't erase your fraction line. Then we're going to put a minus here and a minus here, and we're going to put sin theta and sin theta. Because again, if you had to say negative sin theta times negative sin theta, you would get positive sin squared theta. And if you had to do your foil method properly here, so you had to say that times that, we spoke about.

[00:04:50]Then cos theta times negative sin theta is negative sin theta cos theta.

[00:04:57]And then negative sin theta times cos theta is negative sin theta cos theta.

[00:05:03]And then we already spoke about negative sin theta times negative sin theta is positive sin squared theta, which we already spoke about. If you had to simplify this, negative sin theta cos theta minus sin sin theta cos theta, you would get negative two sin theta cos theta. So, you are basically doing a trinomial here at the bottom. Okay, I know that it's not obvious, especially if you struggle with algebra. This is going to be a little bit of a what the what on earth moment for you, but that's why algebra is so important, and it's important to be able to identify the types of trinomials and things that pop up within trigonometry.

[00:05:39]Then our final step is to simplify. So, I can cancel this one over here with this one over here. So, I'm left with cos theta plus sin theta divided by cos theta minus sin theta. And if you take a look at my right-hand side, let's go back to the beginning.

[00:05:59]There's my right-hand side. Exactly exactly what we got over here. So, we say therefore my left-hand side is equal to my right-hand side. So, what special thing that I implement here?

[00:06:12]I changed one into sin theta cos theta. So, one became this.

[00:06:19]And then, what I did is I did a trinomial at the bottom there once I changed the one into sin squared theta cos squared theta. And remember you're only going to do this if it makes it possible to factorize the expression. So, please don't go moving forward changing all the ones that you can into sin squared theta cos squared theta. Sometimes that's not going to work and it's going to make it way more complicated. So, what I tell my students is every time you see one of these, it will be quite obvious when you need to change it. So, maybe just write this down in your book. You're going to see something like this. Either one minus two sin theta cos theta or you're going to see one plus two sin theta cos theta. When you see one of those, then you know, okay, I must change the one into sin squared theta cos squared theta. Okay, let's try that second more difficult or more advanced identity. This looks kind of simple. So, let's see what's going on here.

[00:07:20]This is an interesting identity in that it looks like not much can be done on both the left-hand side or the right-hand side and that's kind of your giveaway to having to do something special, something different. So, I'm going to start with the left-hand side just to show you.

[00:07:36]And the same goes for the right-hand side that we can't factorize. We can't We can't do anything, yeah? Can't apply a basic identity, really. Nothing's going to work. So, what we need to do is that we And also, I know you're seeing a one maybe just because of the previous example we did and we you think, "Hmm, can't I change that into sin squared theta cos squared theta?" I'm going to show you why this won't work.

[00:08:03]Okay, then you got sin theta.

[00:08:05]You can't actually factorize this because this this trinomial is basically like saying x squared plus y squared plus x, pretty much. You can't factorize that. No, it's not going You're just going to complicate things. It's not going to work. So, we can't change one into the special identity, either. So, what we're going to have to do here, the only only only thing is that we're going to multiply this by one, but not by one, we're going to multiply it by a special version of one. And we've applied a similar sort of thing when we've I had to rationalize denominators, so maybe go back to that algebra chapter because it kind of reminds me of that. What I'm going to do is I'm going to multiply this by one minus sin theta divided by one minus sin theta. Now, the first thing that I want you to notice is this is equal to one. This has a numerical value of one because anything divided by itself is one. Now, the reason I'm choosing to multiply it by one minus sin theta is because this is basically what we call the conjugate of the numerator. So, it is the same term, so one and sin theta, see? One and sin theta, but the sign is different. This has got a positive, this has got a negative sign.

[00:09:29]And I do this for a very very special reason. Look at what happens when I do this.

[00:09:35]But before I do that, I I need you to understand that when you multiply something by one, you're not changing it because anything multiplied by one is itself. So, look what happens when I multiply it by one.

[00:09:46]If I multiply, remember when you multiplying fractions, you multiply numerator times numerator. So, maybe this goes without saying, that times that.

[00:09:55]And you are multiplying denominator times denominator. So, cos theta multiplied by 1 minus sin theta. And what happens at the top here?

[00:10:05]This is going to be It's basically a difference of two squares situation. So, if you had to multiply this out, so that times that, that times that, that times that, and that times that, or you can use your shortcut method, you're going to get 1 minus sin squared theta at the top.

[00:10:25]Here's the working out in red if you are not sure how I got to 1 minus sin squared theta. Then at the bottom, I'm going to leave that the same for now.

[00:10:33]And then I don't know if you recognizing what is going on at the top there. I don't know if you see it, but I hope you see it. I hope you are familiar enough with your basic identities to tell me, "Ma'am, that is a basic identity. I know what that is."

[00:10:46]So, that numerator is equal to cos squared theta over cos theta 1 minus sin theta. Remember when you are proving identities, you're trying to show the left-hand side equals the right-hand side. It is always a good idea to keep your eye on the right-hand side to see how close you are. So, let's go look at our right-hand side quickly.

[00:11:08]Our right-hand side looks like this. Oh, we are so close. This is cos theta divided by 1 minus sin theta. We're almost there.

[00:11:16]All we need to do is simplify this divided by this. So, cos squared theta, remember another way, sometimes this helps some of my students, another way to write cos theta cos squared theta is cos theta times cos theta.

[00:11:30]I'm sure you know that. And then at the bottom I've got one of them. So, cos theta 1 - sin theta. And then what you can do is cancel one of those with one of those. And yep, we are left with cos theta over 1 sin theta. And that is exactly my right hand side. So, we say therefore, left hand side is equal to right hand side.

[00:11:54]Another way to think of simplifying this, it's like if you were to have x squared at the top and x at the bottom.

[00:11:59]You know that you subtract exponents.

[00:12:01]So, you're left with x. Okay, in other words, you're just left with one cos theta at the top.

[00:12:05]Okay, I hope that solving these more complicated identities and was helpful.

[00:12:11]I hope this video was helpful. And I can't wait to see you in another video very soon. Bye, everyone.