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Trigonometric identity example proof involving all the six ratios

Watch as we try to prove a trigonometric identity in real-time. Instead of just showing the clean and final correct path to the proof, we show how we sometimes take a few wrong turns in the path to arriving at the right path. Created by Aanand Srinivas.

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Video transcript

signee + cosecant a the whole squared + cos a plus secant a the whole squared equals or we should show that this is equal to we have to prove this through seven plus seven okay that's that's like a really I don't know it seems like a totally random number over seven plus tan squared a plus cot square a okay nothing immediately strikes me I'm just gonna start seeing what I can do the seven makes me feel like I'm gonna get maybe a pair of a few sets of sine squared theta plus cosine squared a plus cos square a maybe I should get seven of them though to show that I get seven over here let me see one one clue I have over here is that I have sine because he can cause secant away or these are all reciprocals of each other and I have done and caught over here so maybe one approach I can take is to try and write these eventually in terms of this but as always I'm just gonna try and first simplify on this side and see what see what happens I'm just gonna first get rid of cosecant and secant I usually always get rid of them first I don't think it's a good idea but I just do it so I'm gonna have sine square a plus so he's gonna write the sine of a plus 1 by sine of a the whole square plus if I do the same thing over here cos of a plus 1 by cos of a 1 by cos of a the whole square whenever assaulted problems always wonder how they come up with the questions like this is that's that's not part of this videos discussion so sine square a plus one this sine squared a plus one if I multiply the sign over here sine square a plus 1 divided by sine a the whole square plus cos square a plus one divided by cos a the whole square hmm I have a feeling I've made a actually negative progress because I now have if I'd I was kinda hoping to get one minus sine square that would have been useful but one plus sine squared huh I don't really know how that's gonna be useful to me let me think what other identities do I have with me I do know that one plus tan squared is secant squared and one plus cot square is cosecant squared so one thing I can try to do and especially given those two are on the other side is maybe I can try and divide by those in the numerator and denominator by sine squared a maybe cos square a and then I can see what happens I'm just going to try that now I'm gonna try and divide this by cos square a I know it looks a little bit ugly I'll rewrite that in the next step so I'll have to divide this also by cos square a and then I have to divide this also by cos square a that seems to be I'm actually not convinced this is gonna lead me anywhere so I'm gonna pull back on that I'm gonna go back over here and see what could I have done better like my strategy was to convert cosecant into 1 by sine could I have done something else could have divided by something at this stage itself would that have been useful cuz if I divide this by cos theta then I get tan a of cos Arata dani plus 1 by sine e cos a that's what I would have got or cosecant a by cos a hmm that also doesn't seem like it's gonna lead me anywhere they fight done something similar over here and that doesn't seem like it leave me anywhere as well ok another thing I could do is not do any of the simplification but actually just try and expand these out like act like I have no idea what's gonna happen I'm just gonna expand those out and see what happens maybe I'm gonna try that now because I don't like all of these denominator things that are coming in I'm just gonna expand so I'm gonna take all the way from here right and let's let's start doing this so I'm gonna have sine squared a plus cosecant square a plus 2 times sine e in to cosecant a a plus B the whole square is a square plus B square plus 2 a B and this side will come I'm gonna write it down over here let's move down a little bit I have a feeling we're gonna scroll a lot in this video so cos a plus secant a the whole square so that's gonna be the Meuse slightly different color cos square a plus secant square a plus two times cause a into secant a I now understand where the seven comes from I thought it has to have seven pairs of sine square cos square not really we already have four over here because sine and cosecant is gonna their this reciprocals of each other cosine secant a are reciprocals of each other sine square a plus cos square a is one all of these are getting added right so this is going to be equal to one so yeah I actually feel like this is the right I'm beginning to feel the hopeful here this seems like the right approach so I'll get one for these two these two will go in and get a one plus these two will give me a 4 so 1 plus 4 plus cosecant square a plus secant square a in other words I get 1 by sine square a plus 1 by cos square a let's go down even further so we know how he got your so we can in order to remember that so this part of it was a fail attempt so let's go down over here so 1 plus 4 plus this we have let's simplify this one let's simplify this and see what happens so 1 plus 4 which is I'm just gonna write 5 now 5 plus because I'm tempted here to do cos square a plus sine square a but and then let me go look at the right RHS just once for a moment right I mean what is the RHS it has a seven in it hmm and okay I have feeling I know what to do okay so I should not do this I think this is not a good idea because that's this not what the question on the RHS is wanting me to do but it has a 7 there so I'm going to bring two more here but I know how to do that because cosecant square a is 1 plus tan one plus cot square and secant square is 1 plus tan square that's the identity it wants me to use because that's what 0 on the right hand side it doesn't want me to go back to the sine square cos square world so as you notice I have a temptation to convert cosecant and secant to sine and cos all the time it's not always good always useful somewhere cosecant as 1 plus cot square theta and secant square a as 1 plus tan square theta that's right tan square theta the two identities that I that I know and now I have it so it looked like seven why would it be seven but now I understand so five plus one plus one seven plus cot square theta plus tan square theta you've shown it that is equal to the RHS and if you were writing this in a more examing way you'd probably not show all these wrong roots that you took and you may give more reasons for what you're doing and it might slightly look longer but the method is is clear over here oh wait this this is not this is not Tita this is a hey I I can't change the angle right in between that that would I mean even though you probably got the Devoran confused by this but I still think it's it's good to change this cot square theta again cos squared a plus tan squared e and that matches exactly with our RHS seven plus tan square a plus cot square a