hi welcome to another video so in this video what we're going to do is we're going to talk about trigonometric identities super useful super challenging sometimes to prove them and so we're going to start in this video with just an introduction so the basic and basic identities that we have that we're going to use later on to not only improve identities but to make our math a lot easier sometimes we get into this like thought process like teachers are just trying to make our lives harder when they give us identities and it might seem that way because they're challenging but it really will make your lives easier when you get to calc one count two count three and you start dealing with trigonometry and some of these things are very difficult if we don't have a good grasp of identities under our belt and so we're going to start that right now so i'm going to refresh what our identities are even odd the pythagorean identities some of the reciprocal identities we'll use them on just a couple examples right now just trying to get your your brains wrapped around how in the world we structure our thinking to prove identities how do we simplify these things with trigonometry because it sometimes is not very obvious um so we're going to start that right now so some identities that you should have if you have like a little note card man put them on a flash card or something that way you associate the name with the identity that way that when you get there to uh actually using them you're not flipping through your notes all the time also i haven't written them on the board because there's too many of them but there are different ways you can write this like the pythagorean identities we'll talk about in a minute so here's the basic identities that you need to at least be aware of and hopefully know how to use so the reciprocal identities are important they do things like tell you that tangent can be represented as a fraction sine over cosine cotangent can be cosine over sine uh cotangent can also be one over tangent those are all very important cosecant theta is one over sine and secant theta is one over cosine those are all of our reciprocal identities we can write sine and cosine as secant and cosecant but that's typically not what we do i'll give you sort of a heads up about how identities work as soon as we make it through this the pythagorean identities are these guys and things with the squares in them so two squares and a constant usually one so sine squared theta plus cosine squared theta equals one that's true it's a pythagorean identity what i'm not writing up here is things like remember that if this is true i could subtract cosine squared theta from both sides and get sine squared theta equals 1 minus cosine squared theta or likewise cosine squared theta equals 1 minus sine squared theta i'm not writing those but that needs to be i would say obvious to you but at least we can access that we understand that that is that's possible the same thing would happen here so we could have tan squared theta plus 1 equals secant squared theta we could also have secant squared theta minus 1 equals tangent squared theta or we could have secant squared theta minus tangent squared theta equals 1. so any way that we can manipulate that still holds true with these identities and that's kind of nice that means we can use them at least in three different ways same thing with our cotangent squared theta plus one equals cosecant squared theta these are all pythagorean identities but remember that we can change them around to suit us if we need to i'll show you when that happens the last thing are even and odd identities there's only two even trig functions it's cosine and secant the rest are odd which means if i have a negative angle it equals negative of that positive version or in other words opposite inputs give you opposite outputs that happens for sine cosecant tangent and cotangent so sine of negative theta equals negative sine of theta that happens for all of these what we call odd trig functions cosine and secant that's different they're even even means opposite inputs same or equal outputs so if i plug in a positive and a negative angle to cosine i'm going to get the absolute value the same i'm going to get exactly the same output for cosine cosine negative theta is the same thing as cosine positive theta same for secant and so what we're going to do now is we're going to get into just two or three problems of how to simplify some of these trig functions it's going to lead us on the idea of identities like what the world identities do so the two that i want to look at right now are up on the board here's the thought process for how identities work and how simplification works so what we're going to be doing a little while is solving trigonometric identities or i shouldn't say really proving them right now we're just simplifying it's like a baby step towards proving identities here's the thought process number one start with a harder side with identities now there's no sides here because there's no equal sign so you just start with what you have the second step is if you have more than one fraction try to write them on the same denominator so try to combine your fractions we'll see that in a second example after that if you're still stuck write everything as sines and cosines and then use only the identities that you know so the ones that you know are to don't try to make up your own identities it usually doesn't work out all that great um now one thing i will admit is that those those steps i gave you which i'm going to write out in the next video they're not hard and fast rules they're ideas this is general a general idea to get you through some identities are very easy to prove some are very challenging to prove but that set of steps that i'm going to try to build in your head right now really has helped me to prove identities and help me to simplify so let's let's go ahead and start with this one now remember we're not proven identities because there's no equal sign it's not saying this is this now prove it all we're doing is simplifying what we have so tangent theta times cosecant theta the ideas are number one if you have more than one fraction try to combine them try to get one fraction on one side but we don't have that the next idea is if if one of these identities isn't just jumping out at you try to change everything into sines and cosines if you get stuck what do i mean by that well tangent can be written as sine theta over cosine theta and so we're going to do that cosecant theta can be written as one over sine theta and so we would we would try that and the idea is basically put it in things that you're more familiar with most of the time we can change every other function into sines and cosines cotangent does it tangent does it cosecant does it secant does it so we can change everything to sides and cosines and usually we'll have something cancel out it hasn't always happened but this is the reason why we do that so we try to make one fraction you know they have that what i mean is this is two different fractions not things multiplied things added or subtracted try to make one fraction then we're going to try to do if an identity is right there and we use it great that's awesome but if not then start changing things into sines and cosines see if something simplifies see if an identity pops out so right here we have sine theta over cosine theta times one over sine theta well wait a minute we have a sine theta and a sine theta numerator denominator they're going to cancel so 1 times 1 is 1 cosine theta times 1 is cosine theta and then we can leave it like that but it's really more appropriate to write this as a more concise trig function so use some identities that you know so we did was already what wasn't like these two fractions we had to find a common denominator we looked at this and said i don't know an identity with that right now i'm going to write this as sines and cosines so we do that using known identities we can simplify awesome that's just a basic algebra and now we're down to here we see 1 over cosine theta that's great but i'm going to use a known identity to write this more concisely so 1 over cosine theta is let's look right there that would be secant theta i hope you're noticing what we're doing the thought process isn't random the thought process is use stuff that's already here try to write things in sines and cosines simplify them and then also notice that these are two-way streets so we used one as cosecant theta is a reciprocal function one over sine theta but then also we went backwards one over cosine theta is secant theta and that needs to be really clear in your head that identities are not one-way streets they're two-way streets if this then this but also this and this this is necessary it's efficient or if and only if this thing it goes both ways in other words so that's great we simplified this now which one would be easier to use that one that's much nicer if we can simplify this is one reason why we have identities is so that we can simplify things that we get in our math to be easier to work with and that's much nicer let's look at the next one so sine theta plus cosine theta over cosine theta plus cosine theta minus sine theta over sine that looks like just a mess where do you start well if you are trying to prove an identity what would happen is you have an equal sign and this would equal a whole other side and you'd start with what you think is the harder side that's most of the time what you do remember i'm giving you ideas here it doesn't work all the time but generally this is the thought process start with the harder side now we only have one side here so we have to start with that secondly if you have a couple fractions or more than one fraction and they're added or subtracted get a common denominator and make one fraction out of it that's a very good idea last time i'm going to say it these steps don't always work in this order for everything that you do but it's a generally good way to go through and try for identities so we're going to look at this and go yeah let's add those here here's what knots do man i'm i'm begging please don't cross those out it's very common i see it all the time if you've done it it's okay just don't do it again don't cross that up don't cross that out this these are not multiplied together then there are factors you cannot simplify that what you can do is you can get a common denominator and add or subtract those fractions so we're going to do that right now our lcd would be cosine theta times sine theta so we'd say hey those need to have a common denominator this is missing a sine theta this is missing the cosine theta so we're going to multiply by that we also need parentheses right on our numerator because some things are going to distribute we're going to have to combine some like terms so we're going to distribute that don't forget that honor denominator hopefully you can see hopefully you can see that we're going to get sine theta cosine theta and then sine theta cosine theta on our numerator when we distribute we'll get sine squared remember sine times sine is power 2. we write that as sine squared theta and then we get sine theta times cosine theta something very similar here cosine squared theta minus sine theta cosine theta we had to do that because you can't add and subtract fractions if you have a different denominator even if you have trig functions you can't just because you want to make that sine plus cosine or something it doesn't work they have to be the same and so we've done that we said we're missing sine theta put the factor there cosine theta likewise we've distributed we've multiplied here and now we have a common denominator that's great we're going to combine this to one fraction but look at what is going to happen we're going to add our numerators so we'll have sine squared theta plus sine theta cosine theta plus cosine squared theta minus sine theta cosine theta that's going to cancel i'm gonna write the whole thing out so that we see it and it's generally a good step especially oh man i hope you listen right now especially if that's a minus then you'd have to distribute that it would change your signs it doesn't here but in general you put a parenthesis around your numerators so that if that was a minus you could see that sign change and hopefully we see a little clearer here sine squared theta this is our numerator plus plus does not change that so if you distribute a plus or you think about adding both terms you add cosine squared you add negative sine theta cosine theta that would be the term is that you treat that like a negative well if we have positive sine theta cosine theta minus sine theta cosine theta that's gone that's going to completely cancel this was nice because everything was in terms of sines and cosines they're the nicest to work with identities and simplification if if one of these identities isn't popping out like if you have this of course you're not going to want to change that to sines and cosines but in general it's make one fraction make everything sides and cosines if you're not seeing an identity already if then simplify what you can you've just done that and then find an identity that works with what you have left so okay so i've done all that i've made one fraction i've made everything signs and cosines if it wasn't already it was i simplified what i can that's great i've written it more concisely that's awesome that's one fraction and now we try to find an identity if one didn't show up before we have to finally know so sine squared theta plus cosine squared there let's look for something that resembles that now remember it doesn't this i just gave it to you just like a nice softball right there but it doesn't have to be that way i could easily have made that 1 minus cosine squared theta and then we'd say oh yes i can manipulate this that would be sine squared theta that's possible we can do things like that be careful with that use these identities in the the many facets that they have so here we look and say oh that's right here perfect let's go ahead and let's simplify sine squared theta plus cosine squared theta is one so one over sine theta cosine theta the last thing is is sometimes we can outsmart ourselves and go too far uh we can start doing things like well now do i have to find identity like this for that there isn't one that's the last step is only use known identities that way you're not making up stuff now you could do this you could think of this as 1 over sine theta times 1 over cosine theta and change both of those into cosecant theta secant theta that's that's possible so one over sine theta times one over cosine theta remember you can split up denominators by multiplication as long as you multiply the numerator as well so we have that that's one over the same denominator and this is cosecant theta that is secant theta these two identities here and here just going backwards so i'm using the known identities things that i already i already have down in order to simplify this one of these would be most appropriate we don't want to be going any further because there's no identity that makes us any simpler that's really about it we're going to do one more example but hopefully the ideas are starting to build in your head they can start looking really confusing when you start getting them like like that but what do i do i don't know follow the steps and see if it doesn't work for you so that's that's generally how i make it through simplifying and how we start working with identities in just a bit all right last one so we've got this cosine squared theta minus 1 over cosine squared theta minus cosine theta don't cross that out i know it looks very tempting to do but these are not multiplied they're subtracted so we cannot do that we gotta if you have any ability to simplify things it's got to be things multiplied together now that's key that's key to this problem actually so i'm going to model my thing and go through the thought process of how i would approach this i look at it and i start thinking about identities do i see any that are just right there right on the surface that i can use or how it's structured do i have one fraction what's going on so i'm looking at saying it's already one fraction that's fantastic i know i'm supposed to simplify this thing and i don't have to get a common denominator or anything so i'm noticing two things about it number one i do notice that that looks really familiar that looks just like this it's got a cosine squared then it's got a one in fact you could do this now don't don't do it i'll tell you want a minute um but you could subtract one from both sides subtract sine squared theta from both sides and you would get this identity cosine squared theta minus 1 equals negative sine squared theta you could do that now why am i not going to do that this is the second thing i'm noticing everything's already cosines in fact i've got two terms over two terms and everything's already in terms of cosine maybe i can factor this so if i'm trying to simplify remember that factoring is a great way to simplify so when we look at it it's already satisfied the one fraction thing it's already satisfied the everything's in sines and cosines so this is all in terms of cosine well that's the second piece is that if i need to simplify with more than one term factoring creates these products that i can cancel you can't cancel this but if i was to factor it i would be able to do that so sometimes even though you might be able to use identity look at the structure of the problem if it's one fraction and you got everything in terms of cosine see if you can factor it we can't forget that factoring still works on everything that's algebraic and this is a type of algebra it's just that your terms are based on cosine theta treat that it's not a variable and treat that like the variable you would factor so even though it's a function we can do things like look at cosine squared theta minus 1 and think oh yeah that's a difference of squares that's very similar to x squared minus 1. that's the same thing practically as in terms of the structure x squared minus one you'd write this as x minus one and x plus one is two factors this we're gonna write as cosine theta minus one and cosine theta plus one as our two factors on our denominator we've got this cosine theta squared theta minus cosine theta this is very similar to like x squared minus x think gcf think race common factor we can factor that out just because it has cosines in it doesn't mean that your algebra fails it just means you have to really see past it so i'm going to factor out cosine theta if i factor that out remember this is cosine squared theta so factoring cosine theta out we get cosine theta and then cosine theta divided by cosine theta is 1. if you distribute it you can see that cosine squared theta minus cosine theta but this is completely factored and factors great because it multiplies everything together creates these factors that are well that are of course connected by multiplication things that we can simplify if we have common factors on the numerator and denominator so though the whole idea here for like a 20 second recap was it was already one fraction that's great it was already written in terms of sines and cosines in fact it was all cosines and that was the biggest clue to me was that if everything's in terms of cosine i probably don't want to start changing things around if i can factor it now if i couldn't factor it then yeah maybe i'd start using some identities this is one thing i said in the last example was sometimes we can outsmart ourselves so sometimes we look at that and think oh man i've gotta subtract one subtract sine squared theta and use the identity i use the same thing here and then it starts to explode starts getting really really really really big or really really nasty if that's happening there's a chance you could be right and the chance you do it the right way but if it's just exploding and exploding and nothing's cancelling it's getting worse and worse stop stop and go back and see if there was maybe a different way to do it um our better way here would be factoring so if you can factor it try it that's going to create this simplification force you can see right here it does simplify um we can leave it there but there is a couple other things that we can do with this if we write this as two different fractions to simplify this we see a cosine over cosine no you can't just cross them out but you could write this as cosine theta over cosine theta and one over cosine theta that's possible to do this idea of using the common denominator in reverse so this this naturally would give you cosine theta plus one over cosine theta that means you can split this up basically dividing this and that both by cosine theta well cosine theta over cosine theta is one one over cosine theta that's secant theta that's as most as much as we can simplify that that's the most concise that we can write it that's how we simplify things so hopefully it makes sense hopefully you understand this idea of look use the harder side for identity for simplification you only have one side so see if you can write it as one fraction if you have to if there's no identity present then write things as sines and cosines simplify what you can that sometimes means factoring that was already sines and cosines see if you can factor use known identities like we did right here and simplify as much as you possibly can so hopefully it makes sense um thanks for sticking with me next time we'll talk about much more advanced identities we'll start proving them i'll go through these steps again step by step and do lots of examples for you so i'll see you for that