Video: Defining the Pythagorean Identities

Consider the identity sinΒ² πœƒ + cosΒ² πœƒ = 1. We can use this to derive two new identities. First, divide both sides of the identity by sinΒ² πœƒ to find an identity in terms of cot πœƒ and cosec πœƒ. And divide both sides of the identity through by cosΒ² πœƒ to find an identity in terms of tan πœƒ and sec πœƒ.

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Video Transcript

Consider the identity sin squared πœƒ plus cos squared πœƒ equals one. We can use this to derive two new identities. First, divide both sides of the identity by sin squared πœƒ to find an identity in terms of cot πœƒ and cosec πœƒ.

Then there’s a second part to the question that we’ll come on to in a bit. So, in this question, the first thing we’re asked to do is divide our identity by sin squared πœƒ. And when we do that, we get sin squared πœƒ over sin squared πœƒ plus cos squared πœƒ over sin squared πœƒ is equal to one over sin squared πœƒ. Well, the first term is the easiest to deal with, because of we’ve got sin squared πœƒ divided by sin squared πœƒ, we’re just going to have one.

Well, if we take a look at the second term, if we’ve got cos squared πœƒ over sin squared πœƒ, well, we can use one identity to help us here. And that is that sin πœƒ over cos πœƒ, or similarly sin squared πœƒ over cos squared πœƒ, is equal to tan πœƒ, or tan squared πœƒ. But as we can see, this is the other way up to our second term. So, what do we do here?

Well, if we rearrange, we get that cos πœƒ over sin πœƒ is equal to one over tan πœƒ. So, great! But is this useful? Well, not quite yet, because we want our identity in terms of cot πœƒ and cosec πœƒ. And this is neither of these yet. However, cot πœƒ is equal to one over tan πœƒ. So therefore, cot squared πœƒ will be equal to one over tan squared πœƒ. So, that means we have our second term. Great! So, now all we need to do is move on to our last term.

And now to find this term in terms of cot and cosec, we can use one of our other identities. And that is that cosec πœƒ is equal to one over sin πœƒ. So therefore, similarly cosec squared πœƒ is gonna be equal to one over sin squared πœƒ. So therefore, we’ve created one of our new identities. And that one is that one plus cot squared πœƒ is equal to cosec squared πœƒ. Okay, great! So, now let’s move on to the second part of the question.

So, now for the second part of the question, we need to divide both sides of the identity through by cos squared πœƒ to find an identity in terms of tan πœƒ and sec πœƒ.

So, when we do that, we get sin squared πœƒ over cos squared πœƒ plus cos squared πœƒ over cos squared πœƒ equals one over cos square πœƒ. So, once again, we’ll start with the easiest term. And that is cos squared πœƒ over cos squared πœƒ is just gonna be equal to one, because anything divided by itself is just one. So, for the first term, we’re gonna look at one of the identities we looked at earlier, because we’ve got sin squared πœƒ over cos squared πœƒ.

Well, we know that sin πœƒ over cos πœƒ is equal to tan πœƒ. So therefore, sin squared πœƒ over cos squared πœƒ is gonna be equal to tan squared πœƒ. Well, finally, for the last term, we have one over cos squared πœƒ. But we know that sec πœƒ is equal to one over cos πœƒ. So therefore, one over cos squared πœƒ is gonna be equal to sec squared πœƒ.

So, now if we quickly double check, we can see that our identity is in terms of tan πœƒ and sec πœƒ. So, we can say that if we’ve got the identity sin squared πœƒ plus cos squared πœƒ equals one, and we divide both sides through by cos squared πœƒ, we create the identity which is tan squared πœƒ plus one is equal to sec squared πœƒ.

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