so today we're going to focus in on proving identities this is really gonna build off what we learned last class here we go we're going to prove a lot of trigonometric identities today using identities on our formula sheet that we learned about last class so we're gonna do a quick reminder about identities first I'll give you a proving identities cheat sheet that you'll probably want to copy down into your notes and then we're gonna do a metric ton of examples and then I'll give you some practice time today now proving identities is tough at first these absolutely require a lot of independent practice depending on where you're at with yesterday today you can kind of adapt this video lesson for yourself if you want to pause when we get to the examples try them on your own and then watch how I go through them you're more than welcome to do that that would be a really good idea but I am just kind of power through this today so use this lesson how you wish to use it so recall from last class an identity is a mathematical expression that is true for all values of variable or variables in the expression we introduce the reciprocal quotient and Pythagorean identities last class these are all on your formula sheet you can verify an identity is true for certain variable by plugging that variable into both sides of the equation that's kind of silly we've done that several times in the past but whatever proving an identity however this is what we're really focusing in on today means showing that both sides of the expression are identical to each other you've never done anything exactly like this in previous math courses you kind of work similar to it with algebra so there's often a big learning curve with this though again a lot of the times it's just a matter of trying something and seeing where it goes it's not unusual to make a mistake on this and then have to start over that is totally common so here's the cheat sheet you might want to pause and write those six points here down I'll just read through them though you are not allowed to perform operations across the equal sign I did mention this yesterday so in other words you can't do something like subtracting something from both sides for example right when you were proving an identity we're not making the assumption that both sides are actually equal to each other we want to prove that they are the only way you could cross equal sign was if we already knew that they were equal to each other which we don't for being us to prove it so instead you must simply simplify each side of the expression independent of one another so in other words work on the side and work on the right side you can sometimes only work on one or the other sometimes you have to work on both but you're not allowed to cross an equal sign so oftentimes proving identity is a game of tried and hope that works it's not unusual to have to start over when working on a proof here are some tips tip number one begin with a more complex looking side believe me this does help you might be surprised on this but it's actually easier to change something from being complex to something more simple than vice-versa and I know that sounds backwards but believe me on that one if immediately possible so in other words if you see something like right off the bat try using known identities from your formula sheet so if you see something that was on your formula sheet like sine squared theta plus close grid theta just instantly change it to whatever it was equal to to make it simpler if if you can next up this is the one we really focused in on yesterday change all trig ratios into sines and/or cosines this doesn't always work but it's a good rule of thumb and it's a good way of kind of going through it number four this is what's really getting weird look for factoring yea factoring comes back just like it did with quadratic and stuff so the question you should ask yourself is does anything factor out from each piece so in other words does everything on the one side of the equation have a sign in it could you factor the sign out of all of it another one that can come up is does it look like a difference of squares you have something like sine squared minus four or something like that right I'd be different squares you'd be able to square root both those pieces that might help you who knows the big one however we're going to see one like this little later today is it a sum product rule style quadratic and that gets really weird because the X usually in these quadratics now are gonna be a full-blown trig expression so you might have like Coast squared plus three coasts plus five or something like that you know something that you could factor number five here though if there are fractions being added or subtracted write them as a single fraction with a common denominator instead we kind of did that yesterday as well but again you don't like having multiple fractions and let's try to write it as one single fraction I'll help you out and then the last one this is a very rare rule it's really rare that this one comes around it's important to know occasionally you may need to multiply the numerator or denominator of a fraction by its Khan I want to remind you what a conjugate is if you had something like X plus 2 then X minus 2 is its conjugate so it just basically switches a plus and a minus sign around their conjugates from one another we'll see another example like that today as well they're very rare again they don't pop up very often but it is important tool to have in your toolbox all right now here's a good question why do we care why why on earth are we are we doing something like this it's just things we've done it well the real reason is trigonometric functions are used a lot actually in computing right so when people make computer programs including a lot of the ones that would be under a computer right now trigonometric functions are part of that program and of course because your computer is calling on a trigonometric function it asked to perform that function before it moves on now trigonometric functions are very heavy functions for a computer to process it takes a lot of time to process a trigonometric function in your computer if you'd even try this in your graphing calculator if you try to graph like instead of just like going sine x and your graphing calculators you went like sine cosine cosine goes 10/10 sine 10 right and just through a whole bunch in and then did an X inside all of that your graphing calculator is really going to chug it's gonna take a long time to process that so bottom line is what I'm trying to get at the reason we do this is it's nice simplifying these down because if you were ever to use this in a programming application it's important to have it as small and as concise as you possibly can because then the computer program doesn't have to call on trigonometric operations quite as often maybe you don't care about that in btris or whatever it's now I have to do it but there was all right so first we're going to start with a simpler one it's gonna be a verify and then like approve and then stuff kind of question we're gonna consider this expression 1 over Coast X minus cos x equals sine X tan X now the question is saying verify the statement is true for x equals PI over 3 PI over 3 is part of our special triangles so I'm going to do this without even using a calculator PI over 3 is the same thing as saying 60 degrees so you want I'll just do it in degrees just to make my life a little easier here this is gonna be the angle were focused in on and remember the sides of a special triangle will be one half over here and root 3 over 2 over here if we're doing 1 over cosine X cosine of 60 degrees which is the same thing again as PI over 3 is adjacent over hypotenuse so 1 over 1/2 means that that's going to be 2 minus cosine of 60 degrees or cosine of PI over 3 is going to be 1/2 so 2 minus 1/2 there you go equals we want to see if that's equal to sine X 10x well sine of PI over 3 is going to be root 3 over 2 times that by 10 now 10 of this is going to be opposite over adjacent so again that's this divided by this so keep change flip you're actually going to find it that's equal to root 3 over 1 let me solve the left-hand side here first 2 minus 1/2 might as well make this 4 over 2 so they have a common denominator 4 over 2 minus 1 over 2 is 3 over 2 this is going to equal let's see this side root 3 times root 3 well that's just 3 over 2 times 1 is 2 and what do you know left-hand side so the left side equals the right side we've done this correctly it is verified just for this point that wasn't a proof that was just verifying now let's prove it so to prove the the statement is an identity algebraically we have to show that both sides of this are actually equal to one another this is where things get a little weirder remember what I said before when you are proving an identity you are not allowed to cross this equal sign so in other words as much as you may or may not be tempted to do so you can't go and add cosine X to both sides you can't do that we have to assume that it's not equal first and just show that in the same independently so how I usually lay this out is I make a column for left side and I make a column for right side it's easier on a full piece of paper this is kind of a wide screen on this so it might not work too well but we'll give it a try let's just start with one side or the other now it's hard to say which side looks more complicated because remember when we prove one of the first steps we talked about is start with a more complicated side but if I were to pick one that was more complicated here I would say the left side probably looks a little more complicated than the right side so I'll start there 9 times out of 10 you actually have to work on both sides to get it done but still it's important to know figure out whichever side is more complicated first okay so the first thing I'm going to do is I'm going to write this as one single fraction that was another rule that we had in that list oops right back there you go that was another rule we had in that list try to write things action impossible so imagine that is cosine x over 1 to make them the same fraction we even have a common denominator so when a times this by coast x over cossacks so we're going to have 1 over cossacks - coast squared X over coasts and again how I got this was it just times this by Cossacks over Cossacks now that the one fraction I can write it of course or now they have the same denominator I can of course write this as one fraction one minus cosine squared X all over Coast X now there's another thing we can do here you wanna let's put equal signs here and they go through just lay it out there's another thing we could do here remember this cosine squared X is kind of cluing me into that Pythagorean identity we talked about yesterday sine squared X plus Coast one because this is given to you on your formula sheet you are allowed to rearrange this well imagine if I minus cosine squared X from both sides that would give me sine squared x equals 1 minus cosine squared X which is exactly what we have in this actual statement that we're trying to prove so instead of writing 1 minus cos squared X in here I'm gonna change that to sine squared X over a cossack s' now i'm not too sure how much further I can go with this so when I get kind of get to like a point where I am not really sure where else to go you can now switch to your other side and start working on that side instead so now let's think about the sine x times tan X well first of all one of the rules again was turn everything to sign and coast if possible well sine is fine so sine X is just fine but tan we should turn into a sign and kokes remember on your formula sheet is there it says 10 is actually equal to sine over cosine X over cosine single fraction if possible sine X I can just think of that as sine x over 1 multiply these look multiply these through and you're gonna find that sine squared X over cosine we got sine squared X equals cossacks on the left side and side squared over cossacks on the right side we have now shown that the left side and the right side are the same thing so i just like closing those off by saying LS equals RS so therefore it's been proven something else i've never mentioned this before you but in university you do a lot of math proofs if you go and taking university mathematics classes and when you're done of proof there's like just this thing that you usually do to close the whole thing off you have to put one of two things depending on on your mood basically it's up to you when you finish proving something you can put like a little square that you've like shaded in that's just like a symbol and mathematical literature that's like alright yeah I'm done I've proven this but if you want to be really fancy you can write Q II D I don't remember exactly what it stands for it's a Latin phrase basically just means like as it was demonstrated right so in other words QED is basically like a mic drop where you're like boom I've done it proven done right so if you'd like you can put that at the end but at very least I'd like to see when you finish a proof that LS equals RS literally right literally just LS equals RS and maybe put a box around both sides just so I can see very clearly that you've got both sides of the equation equal to each other so because I was able to show the left side in the right side are the same thing that has proven that this is indeed an identity it's not an identity we put on our formula sheet because honestly it looks like garbage but it is an identity nonetheless last question of this here our state the non-permissible values for X in degrees remember a non permissible value is just basically wherever you're dividing by 0 and the only place I'm seeing myself dividing anything is this cossaks right here technically in the 10 you've also got to divide by coasts because I can it's not ever coasts but we'll take care of it right here so what we need to say is Coast X cannot equal 0 let's think about where cosine would be equal to 0 we did this exact same thing last class but I'll just do again you could use your calculator and use the inverse Coast to figure that out but I want to do it the old-fashioned way cosine on a unit circle will spit out your x value so we're looking for the angle that would give us where X is equal to 0 well of course those angles are going to be 90 degrees in 270 degrees because along our x-axis here we can see X is 0 here so at 90 degrees and 270 degrees is where that's going to be at now it wants it doesn't say all of them but we have to assume it wants all of them so what I would say is X cannot equal 90 degrees plus or minus let's see how far apart these are whether nay degrees apart each so plus or minus 180 M where n is an element of the natural numbers there you go that right there would be your answer again these don't come up too often but it is important to know that there are some flaws with this identity it's like there are some times where both sides are just gonna be non-existent if you tried to plug in 90 degrees into this you're you're not going to get an answer that doesn't mean it's not an identity because both sides would have the same flaw but it's still important to note all right we're gonna do a whole gauntlet of like proving questions now so it's up to you depending on where you're at with this you can pause the video and try each one I really would recommend it but if you're really not comfortable with that you can watch how I do each one of these questions and think about that cheat sheet that I gave you earlier in this lesson as to how I go about them remember I'm going through these because I've done these several times before and I feel pretty confident about them so I make usually very little mistakes watch as I end up and I'm gonna stake in this somewhere but that just comes from experience it takes a while to kind of start seeing where we're going to this follow your your cheat sheet and that'll lead you in the way you need to go anyway I'm gonna stop talking here if you want to pause go for it but I'm gonna go over this question prove the following identities remember focus on your left side and your right side do not cross over that equal sign when it hasn't been proven yet very first thing I'm going to do here is because this is an absolute mess I'm gonna change everything that isn't sine or cosine or Coast so I'm gonna get rid of this ten and I'm gonna get rid of this cotangent you can work on the left side and the right side at the same time that is totally okay so let me put my two little columns here left side I'll say is sine X plus instead of tan X I'll write that as sine x over cosine plus coast X on the right side will change this one over cotangent X one over cotangent is ten but we can go better than that we can just say that that's sine x over cosine but little slash down here so we know they're totally divided off that right side is looking pretty simple right now there's not too much more we can do with that so you know what I'm gonna leave that alone we're gonna really like dig in and worry about our left side now so my left side obviously a lot more gated let's use that rule though that says that if you ever see some fractions being added together make them one fraction technically this is sine x over one plus sine x over cosine eads to have cosine as its denominator so I can put a cosine on the top and the bottom of it bottom line is this is going to equal sine X Coast over coast X plus sine x over cosine X whew that's looking like an absolute nightmare but we can add these together and write them as a single fraction just to save myself some room and because I have the advantage of having you know a digital pen here I'm gonna add these together all in one fraction all in one go so if I added those together this would give us plus sine X on the top here all over Coast X now the next thing I'm gonna do here and like don't freak out on this one the next thing I'm gonna do here is I'm gonna factor out a sign off the numerator what that's gonna look like is it's gonna be sine x times the only thing that would be left over in this piece is cos x plus the only thing left over in this piece would be one still all over coast X and then that whole thing over 1 plus Coast X now you might be seeing it you might not be and I won't blame you if you don't I'm gonna get rid of this little division thing here because I'm gonna need some in this room um what you can see now is we have a fraction over something else that isn't a fraction but I can make this bottom piece of fraction by putting it over 1 when you have a fraction divided by a fraction you can keep the first one so sine X cosine over cosine change the divide to a x and flip the other one 1 over 1 plus cos x now hopefully what you're seeing especially for brackets around this is on the top here we have the same thing here as we have done here cossacks plus 1 is the same thing as 1 plus cos x because with addition order does not matter that means when i multiply these together and write it as a single fraction this piece can go and mis piece can go and that leaves us with sine x times 1 which is just sine x over axé and what do you know we have this piece on the left side equal to this piece on the right side so I can close this off by saying I don't know where I have room but I'll put it maybe down this corner here LS equals RS and I don't want to be super fancy I can put like a little box and shade it in that's like the University math way saying boom I've proved this we are done next question prove this identity okay there's one side of this that clearly looks like it's a lot more complicated than the other that would of course be this right-hand side so I'm gonna get started on here LS and RS left side of this equation 1 minus 10 you know what we'll turn that into we'll turn that into a sign and a coast so it's 1 minus sine x over cosine X same kind of idea it's 1 minus cos x over sine X I'm going to still leave that left side alone I'm not going to touch it quite yet I want to I want to work through this remember on the numerator here we have something - a fraction let's write it as a single fraction one is technically cosine over cosine so we could write that now as cosine X minus sine x over cosine here we're subtracting something with sine in the bottom so let's make the sine X over sine X that will give us sine X minus cos x over sine X now this is where it gets a little weird we have a fraction divided by a fraction so let's keep change flip so if we keep the first one cosine X minus sine x over cosine change to at times flip the bottom one sine X over sine X minus cossaks that's what we're left with there now of course we can just keep the top multiplying by the top and the bottom multiplied by the bottom but there's something that I noticed right off the bat that you may or may not have remembered cossaks minus sine X and sine X minus Cossacks are pretty similar to each other but subtraction is a bit different in addition and that order does actually matter but it doesn't matter a huge much because our huge amount of should say because Cossacks minus sine X and sine X minus Oh sex they can cancel out as long as you replace it with a negative one and you can put the negative one anywhere you want on the top you can put it on the bottom eat a piece whatever as long as we need cross these two out it gets replaced by a negative one and if you ever wonder about why that is I can give you like a little example of them corner imagine had 4 minus 1 divided by 1 minus 4 well 4 minus 1 is 3 divided by 1 minus 4 is negative 3 those are the same number it's just one positive and ones negative so you cross it out what are you left with well you're left with negative 1 that's the same idea here okay so basically what I'm trying to say is you're left with on the right hand side here with negative sign x over cosine go with that right hand side let's look at our left side now negative 10 X the writing's on the wall on this one negative 10 X is just equal to negative sine x over cosine is sine over the coast boom we've just finished it right there so you can close it off by saying tell s equals RS put a little thing here you can be fancy with the QED if you want up to you boom we've just proved it that one way or another let's keep going next one again pause if you want to get these ones to try on your own I know this is already gonna be a very long lesson prove this identity both sides here look like an absolute nightmare and it's good like look at this and go where on earth to even start it's already in terms of sine and coasts this is actually going to be those rare ones where we actually have to use the conjugate cool and what's cluing me into that is notice I have a 1 minus sine X here and a 1 plus 9 X here those are conjugates right there and then so when you see some conjugates in here maybe a conjugate method is a way of doing it I deal with a conjugate method what I would do is that just multiply one of my fractions by its conjugate piece so we're not changing what the fraction is equal to so we have to do it on the top and the bottom but we're just gonna change what it looks like so I'm gonna hit the left hand side with a 1 plus sine x on the top and a 1 plus sine X on the bottom so if I multiply that through it's going to be 1 plus sine X on the top over 1 minus sine X times 1 plus sine X on the right hand side we really can't go anywhere with this quite yet so I can't really do anything with this but we will notice is that we can multiply still on the left hand side on that denominator you have our 1 plus sine X on our new multiply this through you might actually be able to tell by looking at its different squares it's gonna become but if you multiply this all through and simplify 1 minus sine squared X now 1 minus sine squared X this at least is a little more helpful because we're not just left with a sine squared we're actually left with a 1 in there as well if you remember if you remember the famous Pythagorean identity sine squared X plus cosine you are allowed to rearrange this because it was given to you on your formula sheet we want to make it look like 1 minus sine squared X so we can subtract sine squared X from both sides and you'll see that Co squared X would equal 1 minus sine squared X that means I can replace this 1 minus sine squared X with a closed squared X so this left side is now equal to 1 plus sine X on the top over Coast's squared X on the bottom and what do you know the right side and the left side are equal to each other I didn't even have to touch the right side on this one let me go so we can say LS equals RS in your little box if you wish do QED these are optional just pry the way really the only thing I need to see is this but in math it's just kind of cool doing it that way at least in my opinion all right anyway here we go next one prove this following identity we're getting close to being done here don't you worry sine X plus cosine X plus secant X equals sine X cos x holy Toledo this one is going to be rough there's a very clear more difficult sign that left hand side is way more intense than that right hand side so let's start there the numerator looks not too bad it's the denominator of that that looks like an absolute nightmare so let me get started with this LS RS toy right now we're hardly gonna need an RS so you know you like put that way over here spoiler alert I know on a big piece of paper it's easier but LS we're gonna leave the top alone sine X plus cosine and X that's the reciprocal signs of 1 over sine X plus secant X which is a reciprocal of coast 1 over cossaks there's two fractions that we're trying to add together remember the rule is when you're trying to add two fractions together get them to have the same denominator and put them as one that's what we're gonna do here so we'll leave the top alone sine X plus cosine EDA times this guy by Kosova Coast and this by signing over sign bottom line is you're gonna have coasts X plus sine x over remember this is all over that original 1 over sine X cos x and again it's sine X close X because I had two times this by Coast and Coast and this 1 by sine and sine so it had sine cos x and sine cos x on both the denominators there so I'll add them together to get cosa X plus 9x cool alright so next thing I'm going to do is we want to divide fractions think of this numerator as sine X plus cosine so ver1 remember the rule is keep change flip so keep the numerator sine X plus cosine change it to a times flip the other one sine X cosine X and technically we're multiplying each piece as a piece here thinking about this now is one jumbo fraction where you have sine X plus cosine X cos x over 1 times X plus sine X notice sine X plus cosine Coast X plus sine X quite the same thing cross those out you are just left with sine X Coast X and boom that was the right hand side as well we didn't even have to touch the right hand side in this it's almost like I knew when I threw it over in the corner up here so we polished off by saying L s equals R s QED smiley face rainbow that's not a rainbow anyway I'm moving on this is I think our second last one here don't you worry we're making better time than I thought it was going to you not know why prove this identity okay that left hand side looks a lot more intense so we'll start with the left hand side secant squared and secant squared X minus 1 okay usually we would change us into sign and coast but I'm going to tell you what I know on the formula sheet that's one there's one that says 1 plus 10 squared x equals secant squared X secant squared X can be replaced with this likewise secant squared X minus 1 could be replaced with just 10 X bear with me on this one okay just bear with me here okay I'm gonna change the secant squared X to 1 plus 10 squared X on the bottom I'm going to change that just to a 10 squared X the right hand side is cosecant squared X I'm gonna leave that alone for now just so I don't have to touch the right hand side again I'll put right side way over in the corner I'm will leave that alone for now you can do it the old-fashioned way you can change the sign and coaster you want to but I'll show you this way I'll be even quicker I'm gonna break my other rule here for a second and I'm actually gonna split apart this fraction in other words I'm going to write this as 1 over 10 squared X plus 10 squared x over 10 squared X so I just took this piece put it over 10 squared X and this piece put it over 10 squared X so I just split that fraction apart the reason I did that is 1 over 10 squared X is cotangent squared X and tan squared x over 10 squared X is 1 well on your formula sheet there is one that says 1 plus cotangent squared equals cosecant squared X that's right on the formula you write it over here 1 plus cotangent squared X equals cosecant squared X that's on your formula sheet that's what this is therefore that becomes this and that was your right hand side so the left side equals the right side once again all right finally last example geez Louise this one's gonna be a bit of a beefcake you might already be able to tell by looking at the denominator on the left hand side that you've got some factoring you got to do here look at this two coasts squared X minus seven cossacks Nanako squared X just Cossacks plus three this right here is like a quadratic you could change these out with some other letter and then solve it the same way as another quadratic and in case you want to what I mean by the quadratic is this is like an adapted some product rule question we're looking for two numbers that have a sum of negative seven and a product of two times three so a product of positive six well right off the bat I can tell you the two numbers that are gonna make that true they're gonna be negative six and negative one we could use that to split apart this middle term I'm just gonna deal with this independently before I even get started here so two Coast's squared X minus six Coast's X minus one cossacks plus three is where we're at this can now be simplified and factor down even further by putting brackets around the first two and then brackets on the second to remember because we had a minus sign here as soon as I put those brackets around here I got to change this plus to a minus to compensate for that because that minus would have to bounce through then I asked myself what can i factor out of this well I can factor out a 2 close so to Cossacks which would leave me with just a cossacks - - to close out of this but leave me with just a 3 - what could I take out of these well there's nothing really I can take out of that so I can just take out a 1 coast X minus 3 notice there's Coase X minus 3 and Cossacks minus 3 this would become coast X minus 3 times to coast X that's this piece minus 1 this right here is the exact same thing as this denominator now let me get started with my proof it's not gonna take too long left side you might know you don't have much room yeah I don't have much room but guess what we're gonna be able to do this no worries left side to coast X minus 1 over the factored version of this that we just found Coast X minus 3 x to coast X minus 1 oh dang look at that too close X minus 1 it's on the top on the bottom but BAM those are gone now this is now equal to 1 over coast is minus 3 bazinga we're done LS equals RS no problem I'll put a square QED I'm getting tired we're done hopefully what you got into this was it's just a game of trying what you know right throw in any of the identities you have try to follow that toolbox you are going to be making mistakes with this that is ok this is something that has quite a long learning curve you'll get it eventually don't worry what clued me in this time was seeing this big chunky piece here very clearly looks like a quadratic to something squared minus something or 7 something plus 3 right that looks like a quadratic to me that was the first place to start these questions are not going to try to trick you up they're just going to be a jumbled mess and it's gonna be up to you to figure out where the pieces go so for practice and this is essential practice page 314 questions 1 to 4 and then question 11 be at 11 see there's a lot of questions on that page 314 that use identities we haven't looked at yet I don't want to be unfair to you and give those to you quite yet because I want you to be using identities we haven't seen we're just using the quotient Pythagorean and reciprocal identities for these ones as always if you have any questions please let me know I know I've gone long on this lesson once again and I apologize but that's just the way it goes again reach out to me if you have any questions best of luck