00:01
So they want us to plug these two substitutions in for x and y, and then simplify this down.
00:08
So let's go ahead and do that.
00:10
But the first thing they tell us is to evaluate this cosine of pi -fourth and sine a pi -fourth.
00:15
So both of these are just going to be root 2 over 2.
00:20
So root 2 over 2, root 2 over 2.
00:25
So now we can factor that out.
00:27
That would be root 2 over 2, capital x minus y.
00:31
And then down here, that's going to be root 2 over 2, capital x plus y.
00:36
So now we can plug these values into there.
00:42
And let's see what that gives us.
00:45
Well, it's going to be root 2 over 2.
00:49
So i'm just going to raise this to the 4th.
00:53
And then we have x minus y raised to the 4th.
00:57
And then plus, let me scoot this down a little bit actually.
01:01
And then plus 6 times.
01:03
So we'd have root 2 over 2, then x minus y squared and then root 2 over 2 squared and then x plus y squared and then plus root 2 over 2 raise to the 4th then x plus y raise to the 4th and then this is equal to 32 alright, so now, actually notice these two also give us square root of two over two raise to the fourth, so let's just cross that out.
01:47
And then square root of two over two, raise to the fourth should be one half, if i'm not mistaken.
01:56
One fourth, i mean.
01:58
So then all of these just become one fourth.
02:08
So let's go ahead and just write that out again really fast.
02:13
Or better yet, i'm going to multiply this entire equation by 4 now.
02:21
All right, because then we clear the fractions.
02:24
So now we have x minus y, raised to the 4th, plus 6x minus y squared, x plus y squared, squared, and then plus x to the, or x plus y, raise to the fourth, and then 32 times 4 is 128.
02:48
All right, so we have that.
02:50
Now, at this point, i don't know if there's any nice way to get anything else to really cancel out.
03:02
We could try to factor out like an x minus y squared and then see what that gives us once we foil everything.
03:10
But i think that might be a little bit too much.
03:14
So it may be best just to start expanding everything at this point.
03:20
So one thing that may help us is the binomial formula.
03:23
So you might recall pascal's triangle.
03:25
So 1 -1 -1 -2 -1 -1 -3 -1.
03:29
1 -3 -1.
03:33
And then let's see.
03:34
So this would be x -0, x -1, x -2, x -3, and then we need one more.
03:39
So 1 -4 -6 -4 -1.
03:43
And we'll talk about what we're going to use this for in a second...