 Greetings. In this screencast, we'll begin our journey of learning how to prove trigonometric identities. In particular, in this screencast, we're going to be working with trigonometric expressions and learning how to rewrite them. And this is the primary tool that we use to prove trigonometric identities. One of the difficulties with trigonometric identities or rewriting trigonometric expressions is that you have to use a good deal of algebra. So you have to bring your algebraic skills to the table. The other thing that we have to use to deal with is that we frequently use previously established trigonometric identities. At this point in the game, we don't have too many of them. We basically have the identities that are listed here, which come from the definitions of the tangent, cotangent, secant, and cosecant. And of course, we have the Pythagorean identity. And because of that one, I frequently like to work with signs and cosines when working with trigonometric expressions. And we'll see how that works with a particular trigonometric expression. So in this case, what we want to do now is try to rewrite the trigonometric expression sine of x times tangent of x plus cosine of x. About the only thing we have at our disposal at this time is the definition of tangent of x as sine of x divided by cosine of x. So we're going to substitute that into the trigonometric expression. And that's what we get. Now, as you look at that now, again, you start looking at each term. And in particular, this one here, you can't do much with cosine of x. But this one, we can multiply this out. And what we get is sine squared of x, sine of x whole thing squared, divided by cosine of x plus cosine of x. And this is where now we have to use some algebra. In particular, what we want to try to do is to take this expression and this expression and combine those into a single fraction. So that's our goal. And what we're going to do to kind of show how this is done is a little bit of practice from algebra. And what we get now is if we have an expression a squared over b plus b, and I want you to look at the similarities between this and this. And you can see the top here is like the a squared cosine of x is like the b. And what we want to do then is combine these into a single fraction. And what we get, of course, we leave the a squared over b. But in order to combine these, we want to have a common denominator. So what we do is rewrite the b as b squared over b. And that gives us a common denominator of b, which allows us to combine these into a single fraction by simply adding the numerators. And we get a squared plus b squared over b. What I would like you to do now is to pause the screencast and see if you can accomplish the same thing down here. In other words, rewrite this expression now into a single fraction. Okay, so it basically is identical to what we did up above. We take the sine squared of x over cosine of x, rewrite cosine of x as a fraction with denominator cosine of x. And in order to accomplish that, we have to use cosine squared of x. Now we can combine those into a single fraction. And we get sine squared of x plus cosine squared of x divided by cosine of x. And now here's where one big difference comes in trigonometric expressions. We actually have an identity that we can use now. In particular, sine squared of x plus cosine squared of x is equal to 1. That's the Pythagorean identity. And so we can finally rewrite this as 1 divided by cosine of x. So actually probably done about as much as we can with this. Here's kind of a summary of what we have done in our work. And again, you can see it's basically exactly what we did. We rewrote the expression cosine of x as a fraction, combined these into a single fraction, and then used the identity sine squared of x plus cosine of squared of x equals 1. And finally ended up with 1 over cosine of x. Actually, what we have proved at this stage are these two identities. What we have up above is this identity here. We could also rewrite that, although I don't see much of an advantage to doing so. We could rewrite that because 1 over cosine of x equals secant of x. So we have now proved our first trigonometric identity. If you recall on a lot of the work we did with triangles and so forth, we often tried to find a way to check our work. And the same is going to be true with identities. There's an awful lot of work that has gone into proving this identity and then some other ones, even more work will be done to establish an identity. And so it's nice to have a way to check our work. One of the things that, when we say this is an identity, let's use this one right here, is that whatever value we substitute in for x, we will get a true numerical equation. The only time that that would not happen is when one of the expressions is not defined, in particular when cosine of x would be zero. What we say is that we will always get a true equation for all values of x for which both sides of the identity are defined. And one way to check that is to use our technology. In particular, what we can do is enter each side of the identity into our calculator, say the left side as a y1 and the right side as a y2 or whatever locations you want to store those and then sketch the graphs. And of course, one of the things with sketching graphs is always choosing a viewing window. Here's an example of what we have there. Basically what I said is, here we graph the left side and we graph the right side. Both sides of this equation, both of these equations, I should say, should produce a graph that looks something like that. This I used the viewing window from minus 2 pi to 2 pi and on the y axis from minus 6 to 6. And that took a little experimentation to get that, to get the graphs to show up well. But since both graphs are identical, that is very strong evidence that this is an identity that our work has been correct. Now one of the drawbacks of this, of course, is that it is sometimes hard to get a suitable viewing window. Our technology has another tool we can use. This actually goes back to a more basic idea of checking. And that's simply to substitute in specific values for x and do the calculation. Do this calculation here and do this calculation here using the same value of x. We should always get the same result. Technically both will be decimal approximations, but they should be the same. One thing we can really do for automating this and get several values of x at once is to use the table building feature of our calculator. In this case, I have built a table for both sides of the identity. I basically started the table at zero and incremented by 0.1. And I just wrote down a few of these here, but you can see what we are getting is the same thing no matter what value of x we substitute. And again, although not a perfect check, that is extremely strong evidence that the work we have done is correct and that this equation here really is a trigonometric identity. So there you have it. We have proven a trigonometric identity. We will be proving more of these in the future. And what we will be doing is building to our list of trigonometric identities that we can use when trying to rewrite an algebraic expression. That's it for now.