>> Here's a situation where factoring is the opportunity, and I see sin squared X here, I see sine squared X here. This is a single term and that's a separate term, so I can pull sine squared X out of those two terms as a common factor, all right? And that's the technique I want to show here. By the way, you might see other manipulations in these problems and we're going to see that an awful lot a little bit later where we start to manipulate with identities. We're going to see a lot of times where we can approach the problem in three or four different ways and arrive at the same kind of result. But here what I want to show is this, that we can take sine squared X out of these two terms. Now when I take sine squared X out of this term, I'm left with secant squared X and when I take sine squared X out of this one I'm left with 1, so we get this. Now how can we simplify from here? Well, of course, I could look at this and try to rewrite this in some other way but. no, I want to look over here. Secant squared X minus 1, that looks vaguely familiar. It might be related to one of the Pythagorean Identities, it kind of looks like something I remember in the past. Well, in order to decide whether this is replaceable using a Pythagorean Identity, what you might do, again, is just go to the sine squared X plus cosine squared X equals 1. Now just divide through by sine squared or cosine squared and see what emerges. Now if I divide -- now, I'm tipped off here that -- I want to have something in here that looks like secant squared, and I know secant squared is 1 over cosine squared, and if I divide through by cosine squared, I'll have, you see 1 over cosine squared here, secant squared. So that's my tip-off that I need to go through here and divide by cosine squared, so I'm doing this.
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You know, a lot of times you'll find yourself doing this mentally. You don't really have to write it down, you can probably see through it in a lot of cases. But at any rate, we get this, and that of course goes out to give 1 and this is tangent squared X and this is 1 and this is secant squared. Ah, well this doesn't look like that. Oh, but wait a second, we can manipulate it into that form. If I subtract 1 on both sides then I find this to be tangent squared X equals secant squared. Oops. Oh, I got it [chuckles]. Minus 1, and so secant squared X minus 1 is equal to tangent squared X, so this secant squared X minus 1 is replaceable by tangent squared X, so that's what I'm going to do...
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...and that's a simplified version. Now, I could take this one step further. This is fairly simple in itself, but I could go a step further and I could think of tangent squared as sine squared over cosine squared, so I could write this.
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And then remember this is over 1, so multiplying here I get sine fourth degree X over cosine squared X.
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