- We are going to see how we can change sine of 4 theta minus sine of 2 theta into a product of sine or cosine. So this is the formula that we're going to use. We see we have the sine minus sine. And then the angles are different. And be sure to look for the correct formula in your book. So let's just get to work. 

We know this is going to be 2 times The first part is going to be sine. So we have sine. And then for the input here, which is going to be the alpha minus beta, alpha is the first angle, which is the 4 theta. So we just put that down, 4 theta. And then we subtract the second angle. So minus the second angle is the 2 theta. So we have 4 theta minus 2 theta. And then be sure you divide it by 2. And this is technically the first angle for the sine. And I will put that in the parentheses. 

And then in the meantime, after that, we are going to multiply by cosine. Alpha plus beta, so that means we have 4 theta plus 2 theta and then over 2, like this. And we do not stop right here. We are going to work out the inside and then simplify complex terms, things like that. 

So this is 2 sine. Well, what's this? 4 theta minus 2 theta, we work out the top first. Combining terms, 4 minus 2 is just 2. And then, of course, that's still the theta. And then we still have the over 2. And then cosine 4 theta plus 2 theta give us 6 theta and then over 2. 

Is this it? No, because right here, this 2 and that 2 can be reduced. And then right here, the 6 over 2 also can be reduced. This is 1. This is 3. So be sure, be sure, be sure you do this. Simplify the answer as much as possible. 

So at the very end, the answer should be 2 sine. And then we just have the theta inside for the first right here, and then we multiply by cosine 3 theta, like this. And I will box this for the answer. 

And let's work out another one, cosine of 2 theta plus cosine of 4 theta. And then we are going to use this formula here, cosine plus cosine. And then the angles are different. And then let's just put this in action. We are going to get 2. And we have the cosine. And then the alpha in this case is the 2 theta. So let's just put that down, 2 theta. And then we add it with the second angle, which is the beta, which is the 4 theta. Let's just put it here. And then over 2. And then we multiply by another cosine. Then we have alpha minus beta, meaning 2 theta minus 4 theta, like this, over 2. And then let's put this in parentheses as well. 

And now what? Work this out. 2 is the 2 on the other side. And we have cosine. And then here we have the 2 theta plus 4 theta. That's 6 theta over 2. And then we multiply by cosine. 2 theta minus 4 theta is negative 2 theta over 2, like this. 

Simplify this. This is 1. This is 3. So the first one is going to be 2 cosine. And we have 3 theta inside. And 2 and 2 can be canceled. And we have cosine of negative theta. 

I want to see if you watch my video or not because you see here, in my previous video, we talk about this. Cosine of negative angle is the same as cosine of the positive angle. So I'm going to just change this real quick. Finally, this is going to be 2 cosine of 3 theta. And then cosine of negative theta is the same as cosine of positive theta. This and that are the same. We'll be using the even or maybe the odd properties for cosine and sine accordingly, just like this one right here. So it should help.