INSTRUCTOR: In this video, we'll be simplifying trigonometric expressions. To simplify a trig expression normally means to write it in terms of one trig function. Notice here, we have our product of three different trig functions, and we want to use our identities below in order to simplify this. And if there's not an obvious substitution, one of the first techniques you can try is to write everything in terms of sines and cosines to see if anything simplifies. 

So we can rewrite this, replacing cotangent x with cosine x over sine x, and we can replace cosecant x with 1 over sine x. Let's go ahead, and try that, and see how it simplifies. So because we'll have fractions, let's go ahead and write sine squared x over 1 times, again, cotangent x is equal to cosine x divided by sine x. And then cosecant x is 1 over sine x. And we can see, now things start to simplify. Sine squared x is equal to sine x times sine x. 

So we have two factors of sine x in the numerator. And we also have two factors of sine x in the denominator. So these simplify out with sine squared x, and we're left with cosine x over 1, or just cosine x. And now this expression is simplified, again, because it's written in terms of one trig function. 

Let's take a look at a second example. Here we have a fraction. And a lot of times, when we see trig functions squared, we can take advantage of these Pythagorean identities here. But I think we'll start by rewriting tangent squared t in terms of sines and cosines. So we'll leave the numerator the same. And when you're simplifying these expressions, there's often more than one way to simplify them. 

Now, again, I'm going to write cosine squared t over 1 since I know I'm going to write cotangent squared t as sine squared t divided by cosine squared t. And notice how two factors of cosine squared simplify out. And now, we're left with cosine squared t minus 1 all over sine squared t. Now things are looking a lot better. 

If we take a look at the Pythagorean identities again, and focus on this first identity, if we were to subtract sine squared theta on both sides, and then subtract 1 on both sides, we would have cosine squared theta minus 1 equals negative sine squared theta. So we can use this form of the identity to replace cosine squared t minus 1 with negative sine squared t. 

Well, negative sine squared t over sine squared t is just going to equal negative 1. And now, we've simplified this trig expression. And you can see it simplified dramatically. We'll take a look at some more examples in the next video.