 In this video, I'm gonna prove some trigonometric identities that involve fractions. Fractions are actually king when it comes to proving trigonometric identities. We often don't like how fractions are hard to add and subtract, which I'll give you that, but they're actually easy to multiply and divide. Cancellation is so much a good thing when you have fractions, so oftentimes we can solve trigonometric functions by making fractions. Consider the following identity we have to prove. Tangent x plus cotangent x equals secant x times cosecant x. So adding together tangent and its complement is the same thing as multiplying together secant and its complement. That's kind of an interesting thing there. We have to pick one of the sides of the equations to work with. I'm gonna pick the left-hand side, mostly because I think addition is more complicated than multiplication when it comes to proving trigonometric identities. So we have tangent of x plus cotangent of x and we wanna make that into a product. The way I'm gonna do that is to make these things fractions. Tangent is the same thing as sine over cosine. Cotangent is the same thing as cosine over sine. All right, that doesn't feel like that's helpful, but put some faith in here. Fractions are good in this situation. We need to find a common denominator to add these together, and so to do that, your least common denominator is just gonna be the product of the two. You're gonna get a cosine x times sine x. Now notice if I were working with the right-hand side a little bit, you can work backwards. That's perfectly fine. If you have secant x times cosecant x, like so, if you wrote those as fractions, secant becomes a one over cosine x and cosecant becomes one over sine x. And so this is the same thing as one over cosine x times sine x. Notice that's the LCD of the sum of these things. So actually adding the fractions is a very good idea. We're keeping our eye on the prize, which we worked a little bit backwards to see that, oh, I want a cosine and sine of the denominator. So I'm gonna times the second fraction. I'm gonna times it by, I need a cosine on top. I need a cosine on bottom. I'm gonna do that on the left-hand side as well. I need a sine on top. I need a sine on bottom. And so this gives us a sine squared x plus cosine squared x over a sine x, cosine x. Because the fractions had its common denominator, the liberative added them together. So how do we combine these things together? Well, we just put an equal sign because, hey, sine squared plus cosine squared is equal to one. That's the Pythagorean identity. So we worked a little bit from the left side, a little bit from the right-hand side, but the key to this one was use fractions. When you wrote these things as fractions, it became much easier to see how to connect these things together. Let's do another example. This one already has fractions here. That's great. I'm gonna choose the more complicated side right here, sine alpha over one plus cosine alpha, alpha's just the Greek letter equivalent of the letter A. One plus cosine alpha over sine alpha, like so. What can I do? Well, we'll start with the left-hand side. And because I already have these fractions in play, sine alpha over one plus cosine alpha, I have a one plus cosine alpha over sine alpha. All right. I need to make this equal to two cosecant of alpha, which, if I take the right-hand side here, two cosecant of alpha. Well, I know that cosecant is one over sine, so this becomes two over sine of alpha. So that's my goal. I want a denominator of sine, which I do have that here, but I have this one plus cosine. I have to somehow get rid of it, but I'm gonna go forward with faith here. I wanna add these fractions together because after all, the right-hand side has no addition. I need to add these fractions together to do that even if I'm a common denominator. The LCD in this situation will be sine of alpha times one plus cosine of alpha. So we're gonna times this one by one plus cosine of alpha. And right here, I get one plus cosine of alpha. And then for the second one, I need to times top and bottom by sine. Now, it's not gonna be much benefit to you to multiply out denominators. It's best to keep denominators factored. So we're gonna have a one plus cosine alpha times sine of alpha, never multiply out a denominator. But in the numerator, it actually is gonna be great advantage to do so. We have a sine squared alpha, and then we have a one plus cosine of alpha squared. So I notice there's a sine squared right here and then there's a one plus cosine, which there is a square inside of there. I feel like I can make some type of Pythagorean identity come out here, but in order to do that, I need to foil out the one plus cosine of alpha squared first. So to do that, all right, what we're gonna get here is we're going to get, foil it out, you're gonna get one times one, which is one, one times cosine, which is cosine. You're gonna get cosine times one, so there's another cosine of alpha. And then you're gonna get a cosine squared of alpha. This all sits above one plus cosine of alpha times sine of alpha, like so. And so some things to note, if we have a cosine squared, excuse me, we have a sine squared, we have a cosine squared, their forces come together and make the number one because it's the Pythagorean identity. So we have a one plus a one, that gives us a two, so I'm just gonna combine that together. We also have a cosine alpha plus cosine alpha, which is a two cosine alpha, like so. And this sits above one plus cosine alpha times a sine of alpha. So now look where we're trying to get towards. We're trying to get towards two over sine alpha. I have the sine alpha and the denominator. I have a two in the numerator. How in the world are you rid of this extra junk? We can't just be like, oh, it magically cancels out and they're equal. This is what we'd say a miracle occurs, right? It's just like, the magic are equal to each other without justification. You would need to show enough justification that a typical trigonometry student could see what you did and understand what's going on there. So we gotta cancel out the one plus cosine alpha and the denominator somehow. Turns out fact through the numerator could be helpful for us because notice there's a factor of two in the numerator. If you factor out the two, you'll end up with a one plus cosine of alpha. Ooh, Bob's your uncle. I can see what we're doing now. One plus cosine of alpha right here times sine of alpha. We now see that the one plus cosines cancel out and we have two over sine alpha which then finishes the identity. We've now proven it. That was a little bit more of a difficult one. I'll give you that. But believe in yourself and let fractions be your ally. They will help you with these trigonometric identities. Let's do one more example in this video. Let's prove the identity one plus sine t over cosine t is equal to cosine t over one minus sine t there. It's like, okay, pick one of the two to go with. It doesn't really matter. You pick the left-hand side or the right-hand side. I'm gonna pick the right-hand side. Again, they look kind of equally complicated to me. The only thing that, the reason I gravitate towards the right-hand side is that a subtraction versus addition, really no big difference there, but having a subtraction in the denominator is worse than having an addition in the numerator. So I say the right-hand side is the more complicated one. Cosine t over one minus sine t. So we have these fractions. Well, I don't have multiple fractions. How am I gonna get a common denominator there? How am I gonna get cosine of the denominator? Well, whenever you see things like one minus sine or one plus sine or one plus cosine, one minus cosine, if you see things like one plus or minus sine of t or one plus or minus cosine of t, I want you to think of the identity that, excuse me, you're gonna a plus b times a minus b is equal to a squared minus b squared. That is to say, I want you to multiply the bottom by its conjugate, switch the sines. So instead of a one minus t, you're gonna get a one plus, excuse me, one minus sine t, you get a one plus sine t. And so you do one plus sine as well on the top. After all, we do want a one plus sine in the numerator, don't we? Now, I fibbed earlier. I said that you don't ever want to multiply out the denominator. There's very few times where you do, and this actually is in that situation where you do want to multiply out the denominator. It takes good judgment to learn these things, right? Or that is to say, judgment that we learn helps us with these things. And the reasons we're being strategic here, one minus sine squared times one plus sine squared, this does multiply out to be one minus sine squared. The reason why we actually do want to multiply out the denominator in the situation, because we can use a Pythagorean identity, that is one minus sine squared becomes cosine squared, right? This is helpful for us because we have a cosine squared, and excuse me, we have a cosine in the top that we don't want, and we need a cosine in the denominator. So if you have a cosine on top, it cancels with one of the cosines in the bottom. So we end up with one plus sine of t over cosine of t. This was the left-hand side, which means we've now finished the proof. And so when working with fractions, you can combine fractions with this nice difference of squares factorization, use conjugates if you have things like one plus or minus sine or one plus or minus cosine. A Pythagorean identity will be useful in that situation.