Hey, everyone. We just became more familiar with our even, odd, and Pythagorean trig identities, and we'll now be asked to simplify some more complicated trig expressions. So, we'll have to use these new identities along with all of those identities that we already know really well in order to fully simplify these expressions. Now, simplifying trig expressions can be tricky because there's not just one way to get to the correct answer, and there are also not specific steps that you can follow in order to get it right every single time. But here, I'm going to break down for you exactly what it means for a trig expression to be fully simplified and then walk you through some strategies to help you get there. So let's go ahead and get started by jumping right into our first example where we're asked to, of course, simplify the expression.

Now, the expression that we're given here is the tangent of negative theta times the cosecant of theta. As we're simplifying these trig expressions, it's important that we know what it means for a trig expression to be fully simplified. There are three things that tell us a trig expression is fully simplified. The first of these is that all of our arguments must be positive. Looking at my expression, I see that I'm taking the tangent of negative theta. So how can we fix that? Well, looking at our strategies, our very first strategy, and probably the most important one, is that we want to be constantly scanning for identities. Since I have the tangent of negative theta, that should tell me that I need to take a look at my even-odd identities, which tell me that the tangent of negative theta is equal to the negative tangent of theta. So using that identity, I can rewrite this to have a positive argument. The tangent of negative theta using that identity becomes the negative tangent of theta. Now my expression is the negative tangent of theta times the cosecant of theta, and now all of my arguments are positive.

Now let's take a look at the two other things that tell us a trig expression is fully simplified. The first is that our expression should contain no fractions, which it currently doesn't. And the next one is that our expression should contain as few trig functions as possible. Looking at my expression, I have two trig functions. I have the tangent and I have the cosecant. So how can we get that to be fewer trig functions? Well, taking a look at our strategies, we are still constantly scanning for identities. And our next strategy tells us that we should break down everything in terms of sine and cosine. Remember, I have two trig functions: the tangent and the cosecant. Taking a look at my identities, I know that the cosecant is equal to 1 over sine and the tangent is equal to sine over cosine. So, I can use those two identities in order to break this down into just terms of sine and cosine. The negative tangent of theta becomes negative sine of theta over the cosine of theta. Then the cosecant of theta is just 1 over the sine of theta. Having broken this down in terms of sine and cosine allows me to cancel some stuff out. Because I have sine on the top and sine on the bottom, that goes away, and I am just left with negative one over the cosine of theta.

So now I have just one trig function. But I do have a fraction now. And remember, to be fully simplified, our expression should contain no fractions. But remember, we are constantly scanning for identities. And looking at my identities, I know that one over the cosine is simply the secant. So this negative one over the cosine that I have in my expression can be simplified down to the negative secant of theta. Now in this expression, all of my arguments are positive, my expression contains no fractions, and it has as few trig functions as possible. So, the tangent of negative theta times the cosecant of theta fully simplifies down to the negative secant of theta. Now, you may have noticed here that we didn't use all of our strategies, and that's totally okay. Sometimes you're not going to need all of them, but we're going to continue using more in order to get where we want to go.

So let's go ahead and take a look at another example and see if we can use some more strategies. In this next example, we are still simplifying the expression, but the expression that we're given here is the sine squared of theta over 1 plus the cosine of theta. Now remember, for this trig expression to be fully simplified, all of our arguments need to be positive, which they already are here. Our expression should contain no fractions, and we want as few trig functions as possible. Now this expression is just one big fraction. So let's figure out how to get rid of that. Now looking at our strategies, remember we are constantly scanning for identities. We want to break down everything in terms of cosine and sine. But this entire expression is already in terms of cosine and sine, so that's not really going to help me. Now taking a look at our fourth strategy here, if we have 1 plus or minus some trig function, we want to multiply both the top and the bottom of our fraction by 1 minus or plus that same trig function. Now looking at my expression, I have 1 plus the cosine of theta. So using that strategy, I want to multiply this by 1 minus the cosine of theta, having the opposite sign of what my original expression had. But remember, if I'm multiplying the bottom by that, I also need to multiply the top as not to change the value of the expression.

Now multiplying this in my numerator, I'm going to leave that factored as the sine squared of theta times 1 minus the cosine of theta. Now in my denominator, I recognize this as being a difference of squares because I have the same two terms, 1 and cosine, just one with a plus and one with a minus. So multiplying this out gives me 1 minus the cosine squared of theta. Now from here, remember we're constantly scanning for identities. And taking a look at my identities here, I know that the sine squared of theta plus the cosine squared of theta is equal to 1. And if I subtract the cosine squared of theta from both sides here, it cancels on that left side, leaving me with exactly what's in the denominator of my expression here, 1 minus the cosine squared of theta. Now I know that 1 minus the cosine squared of theta is equal to the sine squared of theta. So I can replace my denominator. Now in my numerator, I'm keeping that the same for now, the sine_squared of theta times 1 minus the cosine of theta. But in my denominator, I am using that Pythagorean identity in order to replace that with the sine squared of theta. Now I have the sine squared in both the numerator and the denominator, and that cancels out. Now all I'm left with is 1 minus the cosine of theta.

So let's figure out if we need to do anything else here. Here we see that all of our arguments are positive, our expression now contains no fractions, and we have just one trig function, so as few trig functions as possible. We have now successfully fully simplified this down from the sine squared of theta over 1 plus the cosine of theta down to just 1 minus the cosine of theta. Now remember, there are multiple ways to successfully simplify an expression. So, if you want to do this differently, feel free to do whatever works for you. Now that we've seen how to simplify some trig expressions, let's continue practicing together. Thanks for watching, and I'll see you in the next one.