 In this video, I wanna demonstrate how do you disprove a trigonometric identity? That is, how do you show that two trigonometric functions are not equal to each other? For many trig students, they spend so much time proving trigonometric identities. They're proving that these two seemingly different trigonometric expressions are actually the same thing. They sort of get this false impression that every trigonometric expression is equal to each other, because you can always prove them are equal, right? There's so many trig identities, right? No, no, no, no. There are very, very distinct trigonometric expressions. In fact, if I were to cook up two trigonometric expressions randomly, they most likely would not be equal to each other. Two expressions being equal is actually the rarity, even though that's how much time you're focused on, it seems like that's the only thing that ever happens. So how do you show that two expressions are not equal to each other? Well, we actually have to provide a counter example, because saying that two functions are not equal to each other doesn't mean that they never agree on angles. It just means that they don't always agree. There is some angle they disagree with. So let's show that cotangent squared theta plus cosine squared theta is not equal to cotangent squared theta times cosine squared theta. That is the product of cotangent squared times cosine squared is not equal to the sum of cotangent squared plus cosine squared. Why aren't they equal? Now, the idea to find the counter examples, we need to find a specific angle for which the two functions disagree upon. And you might try experimenting, right? Like if you take theta to equal zero, right? The left-hand side, if you plug those in there, you're gonna get cosine, excuse me, cotangent squared of zero plus cosine squared of zero. Cotangent is actually undefined at zero, so this actually does not exist. And the left-hand side as well, if you take cotangent squared of zero times cosine squared of zero, well, again, that also does not exist, right? So the left-hand side doesn't exist. The right-hand side doesn't exist. You're outside the domain of the functions. So this either gives you nothing really. It's like they could be equal, they could not be equal because they're just undefined, right? Clearly, if the left-hand side does not exist and the right-hand side does exist, that means one zero is inside the domain of one function, but not of the other. That definitely would be enough evidence to suggest, but it's like, okay, that didn't work. So let's try something else. So if you tried like, say pi halves, I'm trying to pick easy angles for here. So if they take like pi halves, for example, you're gonna get cotangent of pi halves plus cosine of pi halves, cosine squared of pi halves, right? So cotangent at pi halves, well, cosine I should mention in the situation, cosine at pi halves is zero, cotangent's likewise zero. So you get zero squared plus zero squared, this equals zero, that's the left-hand side, of course. And then the right-hand side, if you take cotangent squared pi halves times cosine squared of pi halves, you're gonna end up with zero squared times zero squared, which is still zero, that's equal to the right-hand side. So that's actually where they agree with each other. So they actually agree at pi halves. Now, let me tell you, this is not a proof that they're not equal, because they actually agree at zero, at zero, at pi halves, excuse me. But this is also not a proof that they're equal, right? Because if all you had to do to prove a trigonometric identity was that provide a specific angle where they agree, that would be a lot easier than how we usually do it. Proving a trigonometric identity suggests that they are equal for all angles theta. But for specific angles, like pi halves, they actually turn out to be the same thing. So we actually, it's like, that one didn't work, that one didn't work, we need to find where they disagree. Let's try pi force. Again, I'm just trying to keep things a little bit easy for us here. If we tried pi force, what happens there? Well, you're gonna get cotangent squared of pi force plus cosine squared of pi force, like so. Well, cotangent at pi force is going to be one, all right? Cause pi force is when cosine and sine are both equal to root two over two. So cotangent and the ratio will be one. So you get one squared. And then for the other one, you're gonna get root two over two squared. This gives us one plus a half, which is equal to three halves. Okay, I can do something with that. Then if we compare the right-hand side, we're gonna have cotangent squared of pi force. And then we're gonna get cosine squared of pi force. So same basic idea. You're gonna get one squared times root two over two squared. So you end up with one times one half, which is equal to one half. And there you go. This actually is an evidence that they're not equal to each other. The left-hand side at pi force is equal to three halves. The right-hand side at pi force is equal to one half. And so we see in fact that they aren't equal to each other. And pi force is not like the exception here. We could have done something like pi sixth, for example, in which case if you had done that one, the left-hand side, if you plug in pi sixth, you'd end up with 15 force, which is not equal to say nine force, which is what the right-hand side would have turned out to be. And so that's also evidence there. So to show that two trigonometric functions are not equal to each other, you just need one counter example. So this right here would be sufficient in this box. So this counter example gives us the evidence that the two are not equal to each other. He was another example, but we don't need, we don't need lots of evidence here. We just need one counter example to show that these two trigonometric functions are not equal to each other.