 In this video, we present the solution to question number 15, for which case we're asked to simplify and graph the function y equals sine x minus cosine x squared on the interval zero to two pi. That interval, of course, is already indicated here on the graph. So in order to graph this, we do want to first simplify it. My first suggestion is to let's take y equals, and then we're going to foil this thing out. So we're going to end up with a sine squared. We're going to get negative two sine x cosine x, and then we're going to get a positive cosine squared like so. Now sine squared plus cosine squared is equal to one, so it's going to give us one minus two sine x cosine x, which it's still a little bit more difficult than we want to do, but the two sine x cosine x by the double angle identity for sine, this actually would change into y equals one minus sine of two x. So that's the function that we want to graph. Notice that this function, because of the plus one right here, there's going to be a shift up by one, and then the amplitude, not the amplitude, the amplitude is going to be, well, amplitude here is going to equal one still, but there is a reflection going on because of that negative sign right there. So we're going to reflect across the midline. So this is a sine wave, so it means we're going to start by going down, then we're going to go up. And then what does this two right here do? It changes the period. So if B is equal to two, that means the period is going to equal two pi over two, so the period is going to be pi. So we have to graph this from zero to two pi, but it turns out there's going to be two cycles on this graph right here. So let's start doing this now. So with the labels provided, we're going to put our midline here at one, like so, y equals one. It's reflected downward, the amplitude didn't change. So we're going to go all the way down. So I'm going to put a little marker right here to indicate that this is one cycle of the graph, zero to pi, and then from pi to two pi, that's going to be a second cycle. So I just need to graph one of these cycles. Sine starts on the midline. It'll return to its midline, of course, at the end of it. It also gets to the midline in the middle. Now normally, sine goes up to its maximum, then back down. But since it's a reflection, we're actually going to go down to the minimum first, and then we're going to get the maximum later on. So if we draw this one cycle, we get something like this. Do make sure that your sinusoidal wave is rounded. Don't be doing some type of like jagged thing. This is no sawtooth going on here. These should be round. It should be smooth when we draw these things. So watch out for that. Don't do that. That would be horrible. And then we're just going to repeat this picture for the next part right here. Draw these little dots to help you connect them. If you want to, we put them together, we get something like this. And this is then the graph of y equals sine x minus cosine x quantity squared from 0 to 2 pi.