 In this video we provide the solution to question number 24 from the practice final exam for math 1060 in which case we have to prove the trigonometric identity two sine squared of theta over two equals sine squared theta over one plus cosine like so. So we have to choose one of the sides of the identity to connect to the other. I'm going to start with the left hand side. I would say that's the more complicated side in this situation. Left hand side is equal to two sine squared theta over two and when you start proving a trigonometric identity you should start off with stating specifically the side you're on. If you don't want to write LHS that's not necessary but you should specifically write out the side whether it's the left right hand side. Some people start doing something like this oh I'm just working working working right and then I connect back so that's that's that's not good good communication. Some people of course are doing these side by side right which of course is not even a valid argument so specifically write out which side you're doing more and I always like to start with LHS to show we're on the left hand side okay. Another important thing that happens when you're proving the trigonometric identity is you must always write out the angle. You can't just say things like sine squared over one plus cosine theta or well I mean if you put in theta that'd be great but if you're missing the angle that's a problem run because this is nonsense as it is right sine and cosine don't really exist by themselves they they need an angle to be talking about. Also like in this trigonometric identity the angles aren't the same you have theta over two versus theta and so part of the process is converting from one to the other and that's actually the reason I'm starting with the left hand side is that I want to apply the half angle identity so which tells us that two sine squared theta over two this is equal to two times well the half angle is plus or minus the square root of one minus cosine theta over two but we're squaring sine so we get something like this so the plus minus and the square root are all going to cancel out and this becomes two times one minus cosine theta over two for which the two cancels there and we end up with one minus cosine theta. Now at this point as you're trying to get one minus cosine to connect you might be not sure what to do right here right because after all we're left with trying to prove this is equal to this and if you look at that trigonometric identity you're like well the right hand side looks more complicated than what I am right now so you might actually sort of just give yourself ample space no one's going to judge you if you give too much space there we could actually work with the right hand side sine squared over one plus cosine like so and see if we can connect it to one minus cosine in some regard and this would change how we go with things it's like again there's a couple ways to do it like if I was working with the left hand side like this one minus cosine theta my strategy would be like well I need a one plus cosine in the denominator but I don't have one so I'm going to times it by one plus cosine theta over one plus cosine theta that's what I would do in which case then the right hand side is going to look like if you fall out the numerator you get one plus cosine theta minus cosine theta and then you're going to get a minus cosine squared theta all over one plus cosine theta for which then when you cancel out the cosines you end up with one minus cosine squared theta over one plus cosine theta like so for which case then we're like oh you draw a really long equal sign because those are now equal to each other right one minus cosine squared is equal to sine squared and so honestly if I was proving this identity that's how I would have done it trying to match up the denominators and go from there but maybe you didn't see that right what is an alternate approach you could do to one like this what if you're trying to work with the right hand side well look at the right hand side it's like I need a one plus cosine of the denominator I have a sine squared I have to get some how to get rid of the sines and just end up with cosines in which case then the Pythagorean identity comes into play here it's like maybe I rewrite sine squared as one minus cosine squared theta over one plus cosine why would that be useful because one minus cosine squared has a difference of squares factors this would factor as one minus cosine squared excuse me one minus cosine times one plus cosine all over one plus cosine for which the one plus cosines cancel out and you're left with one minus cosine theta so put an equal sign there to connect those things together so again there's and if you want to do a long equal sign you can do that it doesn't matter so again there there's kind of two ways I might approach this one and whatever you do is it's fine right what the two approaches are equivalent they're both acceptable no big deal so the thing is if you get stuck in the middle it's okay move to the other side and so some important things to remember here is when you're working with a when you're trying to prove a trigonometric identity you start with the left hand side and end with the right hand side or start with the right hand side and end with the left hand side it doesn't matter each statement needs to be connected to each other with inequality an equal sign because that's what we're saying we're saying this is equal to that this is equal to that this is equal to that to that to that to that until by transitivity all of these things are connected to each other and each jump each equal sign shouldn't be a huge jump like this one right here was just stating what the left hand side was this was using the half angle identity if this is one of the standard identities that's on our formula sheet like the basic identities half angle double angle you don't need to cite them you can just use them because we'll know the half angle identity and then the next one just an algebraic simplification an algebraic simplification working when this one backwards we use the Pythagorean identity we algebraic factorization and then algebraic cancellation every equal sign is justified and they should be equal signs there shouldn't there should be an equal sign if there's just a gap your I'm supposed to infer that those are equal no you should be explicit here we also should avoid things like arrows of some kind because that suggests some type of logical implication which it's not a logical implication it's an equality like these two things are equal to each other it's not like all dogs are animals and all animals are alive therefore all dogs are alive you know it's not a logical implication it's an equality so we should be using equal signs as well always make sure you list the angle it's very important we list the angle and if you follow those steps you'll get full marks on a trigonometric proof like this one