7.4 advanced functions is proving trigonometric identities now some people find this really hard some people really like them I think the more you do the more you will like them I know that sounds kind of crazy but it gets to be kind of a puzzle so I'm going to go over a couple of examples and then I'm gonna make this lesson fairly short but I will follow it up with a second lesson proving a lot more different identities so if you have any particular ones from your homework assignment that you're stuck on and I haven't yet done the lesson - I know some of you are already watching these lessons as I post them so if you have specific questions you'd like you can send me a little message at the end of this lesson okay so let's go over some of the rules that you should be using when you're trying to prove identities the first thing is to check to see which side is more complicated now this holds true really well for the grade 11 curriculum but in grade 12 yeah it's still you're still looking for the most complicated side but at the same time you often have to work with both sides of the equation to simplify them before you see something that's similar work each side separately do not move across the equal sign in other words you're going to say the left side is equal to this and work with up until you get an answer and right side break down into sines and cosines this is a really key point if you can break it into these little building blocks you're more likely to see some sort of similarity often you're asked to find a common denominator not asked you but you will need to find a common denominator this is common oh isn't that funny common when you have two terms on one side and only one on the other in other words how are you going to get from having two parts to one part well obviously you're gonna have to somehow add or subtract them together to get this one part sometimes you need to factor oh yeah someone difference of cubes difference of squares and sometimes just a basic train old meal you should use substitutions like sign to weigh that you've learned or close to a sometimes you have to expand like the sine of X plus y you might have to expand that using all these little tools that we've learned in the last couple of sections Coast squared a plus sine squared equals one is often used the variations on this you might be replacing one with this or replacing this with a one and you'll go oh I was so easy why didn't I see it so watch out for that one and sometimes you need to multiply by a conjugate be creative don't give up you will get better with practice okay so let's go we're going to do three or four examples and then like I said I will dig up some of the more difficult ones and do them for you in a follow-up lesson so I have sine X tan X equals secant X minus cos x well they both have something that I can break down this one is all one part this is two part so I'm thinking of those things as I do them and don't forget to look from left to right as you're working with it to see what direction you should be going in so I'm going to write what the left side is what the right side is and then I'm going to start simplifying okay so I know that tan X is sine x over cosine so I'm going to replace that always replace Tim or a reciprocal function by this sine and cosine so now I have sine x times sine X oh that would just be sine squared X isn't it over close X now let's just run over to the right side here for a minute to see what we've got here we have a minus sign here so what's secant secant is 1 over cosine 1 over cossacks - coast x so i see how i have only one term on this side and i have two things here so that means i'm going to find a common denominator here which of course is going to be cosine X going to x cossacks over cossacks whoops coast x and that's going to give me 1 minus rho square X all over Cossacks and OG isn't 1 minus cos squared X the same as sine squared X so there we go I don't know what your teacher asks you to write once you've finished in my class I would have had them to say left side equals right side and you can either rewrite this saying that they're equal I also used to get my students to write QED which means quod erat demonstrandum therefore it has been proven maybe you wanna see if you teach will let you get away with that quick little short form okay so let's flip over and look at something a little more complicated so here I have 1 plus coconuts tan y equals sine look X plus y well we just learned how to do that so you can be able to break that up and well let's just start with the left side so I have 1 plus cotangent and X sorry 10 Y and I know that coat an X that break it into sines and cosines right so co'tin is cose x over sine X and tan Y and it's multiplied here so I have sine Y over coast why okay now when I'm at this point I just keep looking over here and notice how this denominator is already sine X cosine which I would have here right I just multiplied these so I've closed X sine Y over sine X cosine and I have a 1 so I mean you're going to do something with that one or I'm going to I'm well what I'm going to do now is I'm just going to lead this side and run over here because the right side was pretty complicated with this addition formula so sine of X plus y remember is sine X Coast y plus Coast X sine Y and that's all over sine X Coast y so now I'm going to take a look at the other side here I'm just going to write this like this first close X sine Y over sine X Coast I am I'm going to compare what's the same here well I've got the right denominator but I have this one here and I have this so look that's like this and this one over this isn't one right so I have what I need here let me just show you how I got that in case you see how did you make a 1 out of that I know some of you are saying that I can hear you ok so if I had something like this and always I always tell my students to think of a really basic simple equation if you're not sure what to do when I have something added together over a common denominator it's like me saying if I had two plus three over four okay so say this was - this would say this was four that's the same as two over four plus three over four so that's what I've done here I've broken this so I have this one which becomes the one and I have this one which is still this so now we have left side equals right side or left side equals right side QED and that's always nice to write after it you will have to face okay let's go down let's do this one here Co squared theta minus sine squared theta whole square theta plus sine theta cos it equals one minus tan theta obviously this is a more difficult one but this tan can be broken down into sines and cosines because we have lots of those here that's probably what we're going to need to do so with the left side here I can see that these all some factoring skills that I could use for them right co squared minus sine squared that's a difference of squares so I'm going to say Co stata plus sine theta times cos theta minus sine you know I took up more space than I wanted to and in the denominator I have a common factor right I can take out a Coast theater what am I left with Coast theta plus sine theta now all these things are multiplied together right and once things are multiplied together you can cancel them out because little packages divide into each other and that leaves me with Coase theta minus sine theta over coast no now you couldn't go over to the right side here and break this down but I'm looking at this and I already see that answer do you because look ko stayed over coasted and that's one and sine theta over Costa is tan theta so I would just say oh this is equal to one minus tan theta and right side equals one minus tan theta so therefore left side equals right side and a little tricky thing if you get to here okay and sometimes don't don't tell you teacher I told you this but if you just jump and say oh that's equal to this even if you're not sure and say left side right side she's gonna think you knew that but you should write it down right okay last one sine 2 X 1 minus Coast 2 x equals 2 cosecant 2x minus tan X that one looks really complicated doesn't it okay so let's see what can we do with the left side left side sine of 2x well the sine of 2x is 2 sine X cos x and I'm all over 1 - coast to X I'm gonna write that down here anyway even though I know I'm going to want to change this coasts of 2x let's go to the right side just to see what kind of denominator I want because as you know the cause of 2x has 3 possible equations that I can substitute in here so I'm going to go to the right side first and I put it right over way over here so to kick cosecant 2x that's 2 over cosecant 2x is 1 over the sine of 2x so I have 2 over sine 2x and the tan X is going to be sine x over cosine identity for sine of 2x and that's 2 sine X cosine so I'm going to write that here 2 sine X Coast's - sine x over cosine cyl into each other and I have a 1 up here and to subtract these from each other I need a common denominator so I'm going to write this as 1 over sine X cos x - now that means I have to multiply this side by sine X over sine X right common denominator so if I do that I'm going to have sine squared X sine x times sine x over sine X coasts and that means this one's going really well isn't it and it's easy to see what you can do with this I have 1 minus sine squared X over sine X Coast and what's 1 minus sine squared X well that's just Coast squared X isn't it 1 minus sine squared X remember we had sine squared X plus Coast where x equals 1 so one mine sine squared X is post squared X so I have Co squared X sometimes they just flow so nicely don't think you're probably going come on the set up this is painful okay so I have a co squared over at Coast that's gonna get rid of the squared and I have post x over sine X okay now you could say well that's just coat an X but we're not going to go there because we've got it nicely into sines and cosines okay so if you look here now you see my denominator is sine X and I have coast to X here so remember that one of the formulas you could use for coasts to X that was 1 minus 2 sine squared X and that will bring in a sine X into the denominator which is what I really want to have like I do here because I want I want to get this to be Coase over sine so I'm going to substitute the coast to X with this don't forget the whole part and don't forget this minus sign remember most of the mistakes and math are made because of negatives something under the paper there okay so 1 minus coast 2 X is going to be 1 minus put a big bracket here okay that way you won't make a mistake 2 sine squared X okay we're moving so I've 2 sine X coasts over 1 minus 1 I already do it all up 1 minus 1 plus 2 sine squared X so now I have 2 sine x coast get tired over 1 minus 1 that's gone so I have 2 sine squared X whoo look at that eye so I have 2 & 2 1 sine X with 1 sine X and bingo I've got the same as a head on the other side it was a long one left side equals right side Q II D and I hope that made you very happy okay so like I said if there's a specific one you'd like me to do get to me in the next few days it's um I've got a lot of company coming tomorrow so I won't be making a lesson for a couple of days okay all the best