 In this video, we provide the solution to question number 21 from the practice final exam for math 1060, in which case we have to prove the trigonometric identity cosine of x over one plus sine of x minus one minus sine of x over cosine of x is equal to zero. All right? So I would definitely say that the left-hand side is the more challenging, the more complicated side. I always like to start with the more complicated side if possible. I think it's easier to simplify than to complicate a mathematical expression. So our proof, we're gonna start with the left-hand side here and then we're just gonna rewrite exactly what we see there. Cosine of x over one plus sine of x minus one minus sine of x over cosine of x. It's a good idea to actually write it out, right? Because after all, the trigonometric expression, excuse me, trigonometric identity proof is supposed to show that the left-hand side is equal to the right-hand side. So you need to start with the left-hand side. If you actually wanna write LHS, like I do, that's a great idea but not actually necessary, you do need to start with this. Too many people do things like, oh, this equals and they go from there. That's not proper notation, that's not proper. You can lose points from formatting this thing incorrectly because a proof is a sequence of arguments that do need to be communicated correctly. Now, if you're wondering why I put a huge gap in between here, this is actually because I'm getting ready for what I'm trying to do next year. How in the world is this gonna combine together to give me zero? Well, I'm subtracting stuff, so maybe if I combine the fractions together, everything cancels out. That's my goal here, but they have different denominators. So in order to add them together, they need to have a common denominator. So if I times the first one by the denominator of cosine, I have to do the same thing to the top. And then for the second one, you're gonna get one plus sine x, one plus sine x, like so. So that allows us to have a common denominator, we can add these things together. So in the numerator, we're gonna get a cosine squared when you multiply that out. Subtract from the second one, you get one minus sine x times one plus sine x. We'll do that one in just a second. And then when it comes to denominators, don't multiply them out, leave them factored. You're not gonna do yourself any favors by multiplying it out. So we get one plus sine x times cosine of x, like so. Now, in the numerator, we have the one minus sine and the one plus sine. We should foil that thing out, in which case we're gonna get cosine squared minus, and this all here is being subtracted. You get one times one, we're gonna get a sine x, one times x, then you're gonna get a minus sine x, because you get negative sine times one, and then you're gonna get a minus sine squared, like so. And this all sits above the denominator cosine of x, one plus sine x, we're not gonna multiply that out, okay? And so then we need to continue to multiply out the numerator. Note, well, I should say simplify the numerator. We have a sine x minus sine x right there, those cancel out, right? And so in the numerator, we end up with a cosine squared x, we get a minus one, and then we're gonna get a plus sine squared, like so. And this again sits over the denominator cosine of x times one plus sine x, like so. All right, and so now we're like, how is this gonna cancel out? Well, I have a cosine squared plus sine squared. That's equal to one by the Pythagorean identity. If you wanna list the identity on the page to make the greater realize I'm using the Pythagorean identity, that's perfectly fine, but the elementary identities can be used without citation because they're so fundamental. And so cosine squared plus sine squared is equal to one. You're gonna get one minus one over cosine x times one plus sine x, like so. Then the numerator simplifies to be zero over cosine x times one plus sine x. And whenever your numerator is zero, that's gonna make the whole fraction zero and that's equal to the right-hand side. So we've now proven this trigonometric identity. So some things that I do need to point out when you're proving a trigonometric identity, particularly formatting issues, because again, proving an identity is just as much an exercise of communication as it is of mathematical logic. So you do need to start with the left-hand side and end with the right-hand side or vice versa. You can start with the right-hand side and end with the left-hand side, that's fine. But in this sequence of equalities, the left-hand side needs to be included either at the start at the end and the right-hand side needs to be included, again, at the end or at the start, depending on which direction you're going. It doesn't matter which direction, just you do have to go from one to the other. And you have to, it's a sequence of equalities. So each statement should be connected to each other by an equal sign. So equal sign, equal sign, equal sign, equal sign, equal sign. You know, you have all these equal signs that connect each expression together. If those equal signs are missing, then you haven't proven a trigonometric identity. Also, some students like to draw things like arrows, like this, that's not an equal sign. So this arrow would suggest that, oh, we're progressing through an argument, a calculation of some kind, but we're not calculating anything right now. We're proving that two things are equal. They do need to be connected by equal signs. The missing of equal signs are the substitution of something else for an equal sign is not proper notation, it's not proper communication. And so there are potential demerits there if we do not connect these things with equal signs. So we should start with one side of the equation and connect it through a sequence of equal signs till we get to the end. Now, each subsequent equality, it shouldn't be a major jump, right? Like as you went from the first to the second, all we did is we showed you as we changed the denominators, great. Then for the next one is just some algebraic stuff, like foiling and distributing things like that. And then from like here to here, we just use a single trigonometric identity, the Pythagorean identity. If you're using a fundamental identity or basically any of the identities from the formula sheet, you don't need to justify those. You can just connect them with an equality because we should know those identities and that explains it. So each connection should be small. If we take two big of jumps, then it seems to suggest that you're either doing, you're not showing enough or maybe you're hoping that the two things are the case. You often see the trigonometric proof that looks like left-hand side equals a miracle occurs, right? There's no logic. There's no sense to why this happens. But you know, things equal to each other. What happened? I don't know. So it's very important that we express our trigonometric identities properly to get full credit on a question like this.