>> Let's look at some other techniques. We have here 2 minus secant squared X equals this. Now I mentioned earlier that generally speaking we start with the more complicated of the two sides. Well it's kind of difficult to tell which side is more complicated in this particular problem. But I'm going to pick on the left side of this equation, this identity, and I'm going to try to make this expression look like that one. And I'm just going to manipulate over here, until this looks like that, okay? And so what I'll do is I'll think well let's see, I'm trying to, and I'm going to look over here kind of as a reference. I'm going to think to myself, well I want to write this in terms of tangent squared. Hmm, well I can do that by recalling one of the identities, one of the Pythagorean identities is that tangent squared X plus one is equal to secant squared X. So this can be replaced with tangent squared X plus 1. You see? It goes like that. And now we just kind of follow our nose through it. Let's see, we have 2 minus, oops don't need that. Tangent squared X minus 1, and then we have 2 minus 1 is 1, so this is 1 minus tangent squared X, and oh happy day these now look the same, and so I'll write 1 minus tangent squared X on this side. Now a lot of times there are many manipulations that work equally well in situations like this. For example, I could have decided up here well instead of replacing secant squared X with tangent squared X plus 1 from that identity, I could have done this, I could have said well, why don't I take another approach and just kind of separate the two into 1 plus 1 minus secant squared X, you see, and think about these two together, and now perform a manipulation. And now remember, its tangent squared X plus 1 equals secant squared X. And so I can kind of manipulate this by thinking of this as oh let's see, it's 1 minus secant squared X equals minus tangent squared X, you see, and then make this kind of substitution right here, and then lead to the same position where I was before. It is important though in making the various substitutions that are involved here, to make sure that someone reading the problem can follow the various steps to that logical conclusion. We know that we have equality here. And the tendency on the part of students is to get to a point where we kind of see that the two sides are equal, and then skip a couple of steps toward the end. Oh, I can see through that, and therefore this is equal to that, you see? But it's very important to show all of the steps in that whole process.