>> We've talked about the difference between expressions and equations before. I'd like to revisit that a little bit because I wanna concentrate a little bit on a special kind of equation that we're dealing with here in this chapter. We're dealing with identities and that's a special kind of equation. Now expressions are like this, you know, previously. You notice no equal sign and in trigonometry this is an expression. Now with expressions, generally we either evaluate them if we're given a value for a letter or we simply manipulate them, we generally simplify them. Well equations have equal signs involved in them and there are two kinds of equations, one is the conditional equation like this from our previous experience and in this circumstance we're trying to find the value or values of whatever letter may appear that'll make the statement true and here in trigonometry, we're gonna be solving equation like this. Right now, though, we're concerned with identities and identities we've seen before but they've manifested themselves as something that look like this. Now an identity is characterized by the idea that it's true for all real numbers. So for all real number replacements for X, this statement of equality is true and you may recognize this as simply the statement that corresponds with the distributive property. Now in trigonometry, this would be an identity. It's true for all real numbers, all real number replacements for X or all angles X. Now our idea here in trigonometry is to actually verify identities. We're going to be given a number of identities and we're going to verify that they are indeed identities by going through a certain prescribed process. And that process involves manipulating one side of the equation or the other and sometimes both but generally just one side or the other into the form of the other side and usually we start with the more complicated side. Here's what I'm talking about. Suppose we wanna verify this identity. Now here's the one we saw a little bit--just a moment ago as the example. But to show that this is an identity or to prove that this indeed an identity, we take one side of the equation, this side let's say, and we manipulate it a little bit with substitutions and so forth and cause it to look like that side. Here's how it goes. Tangent X we know to be sine over cosine. So it's sine X over cosine X. And cotangent X is the reciprocal of tangent X. So it's cosine X over sine X and now we get cancellations all around and therefore 1 equal to 1. Now often in working with identities in this way, we're not even gonna write the item on the other side of the equation. We're gonna just manipulate on one side and then in the last step we say that these two items are equal to one another, and this one is pretty obvious. We know that these are reciprocals of one another and when we multiply reciprocals, we get 1 as their result.