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23. Prove the tac b(oz8 Zcuan4} 1csc an is 2 identity 40 0 2 4...

Question

23. Prove the tac b(oz8 Zcuan4} 1csc an is 2 identity 40 0 2 4

23. Prove the tac b(oz8 Zcuan4} 1csc an is 2 identity 40 0 2 4



Answers

$23-40$ . Prove the identity.
$$
\cos (x+y)+\cos (x-y)=2 \cos x \cos y
$$

All right. So I'm just gonna look at the left inside and transform it into. So first, I'm given sign of in addition formulas. So let's rewrite that sign of the first angle Times co sign of the second angle plus sign, uh, excusing co sign of the first angle time. Sign of the second angle. Okay, cool. That doesn't tell me much it. Now I'm gonna do the second sign. Its attraction formula. So I'm gonna do sign the first angle Times co sign of the second angle minus sign. Excuse because of the first angle times Sign of a second angle. Okay, But I'm subtracting these two values, so I'm distributing the subtraction sign to both. Now he's here, so it's gonna be minus sign of x co Sign of why two negatives make positive plus co sign of X. Sign of why? So when I combine these two sign of Exco scent of y minus end of excuse and why it's like one minus one thes will cancel and then co Sativex sign of y plus cosign of extent of eyes like one plus one to cosign of X. Just upset And why? And that will be the same Miss my great inside

So we want to prove this identity. So cosine of X plus y times the cosine of X minus y is equal to cosign squared of ax minus sine squared of Why? So why don't we try to use their some of angles? Identity? Here we know we take coastline of the first times, the cosine of the second minus the sign of the first times the sign of the second. And for this side, we know that we have or for that function, we have cosine of the first times, the coastline of the second. And we have plus sign of the first time sign of the second. And let's see if we can get that to equal this. So why don't we do some multiplication? So we multiply these two together we get cosine squared of acts times cosine squared of why, and distributing Here, we're going to end up getting plus Kassian ax cosign y I'm sign of acts. Sign away. It looks like a mess right now, and now we'll distribute this through and that gives us lo and behold. It gives us negative cosine x cosine y sine x sign y. And then we have finally right here to distribute that. So we have minus sine squared of acts and then, uh, sine squared away I and so we can see and that equals. We're hoping to prove the other side. So we know that these dropout and so we have on the left side get back to black. I see a cosine squared of acts and ah, cosine squared of why and minus a sine squared of X and a sine squared of lie. And I'm looking ahead to what I want to prove And I'm going Oh, goodness. I would love to have this guy like drop out and I'd love to have this guy drop out and then move on my merry way. So we're gonna have to do some substituting. It's not going to just fix itself. So how about I don't want that to be there, So I'm going to substitute Pythagorean identity and I'm going to substitute in place of cosine squared of why one minus sine squared of Why? Let's see what happens if we do that. And likewise, I can see I don't want this there. So let's see if by chance, if I distribute here, I'll get that coastlines quarterbacks like I want and I'm not going to get anything to cancel out. So I've got to do something to get rid of of this. So I'm going to substitute in place of science squared one minus cosine squared of X Not supposed to be a cosign script that doesn't look very good there. So we'll race that just trying to get something so that I can get rid of that term that I don't want. Okay? And so I know that those air justified substitution. So there's going back to reiterate. Used are some of angle identity for Kassian used our difference of angle identity for Kassian. I multiplied things out. Two things canceled. That was good. But I'm left with this and I can't get anywhere yet. I want it just to be a cosine squared off x minus, a sine squared of why? So I'm going to get rid of some things. So when I distribute here, I have cosine squared of acts minus and I have cosine squared of ax times, the sine squared of why. And here when I distribute, I get minus one and I get plus oops minus one sine squared of X. I'm sorry. Science whereto Why? Well, I kind of think of this. I'm distributing this negative through Thio here, and I'm also distributing the y squared at the same time. So I'm distributing this and I'm running the negative through. So the negative times the sine squared away, I I got this. And then the negative times the negative will be positive. And I have cosine squared of x and I have signed squared of why Oh, my gosh. Look what just happened? These are going to cancel out. And now I have proven that cosine squared of acts minus sine squared of Why is equivalent to my original statement Now your teacher may have you write down the reasons and so on why you could do what you dio. That depends on the instructor. So you should make sure you find out whether you have toe put down justifiable reasons next to your steps. I require my students to do that. But you see what your teacher ass

Okay, So what I'm gonna do is I'm just gonna focus on one side, simplify as much as possible, and then focus on the other side's by as much as possible. So first, I'm gonna focus on just the left hand side. Sign of Piper to mice X. This is this assigns attraction formula. So applying the formula Sign of the first angle Times co sign of the next angle minus CO. Sign of the first angle time. Sign of the next angle sign of pi Over two is the Y value at 90 degrees, which is gonna be one. So this will just be, um one Times CO sign of X minus, uh, co Cenepa River Tuesday. Expel you, uh, at 90 degrees, we're just gonna be zero. So this is just gonna be zero time son of ex, and this will cancel over here. So my left hand side is just gonna be Cosette Rex, my right hand side. I have a sign addition formula. So I'm gonna do sign of the first angle Times co sign of the second angle, plus, um, co sign of the first angle time. Sign of the second angle. Okay, Sign of pi over two, like we said before is one times co son of ex and then co Santa Pirate to remember is zero times sign of X. So this will cancel. And then my right hand side will also be cosign of X to my left hand side of my right hand side er equal. So therefore, we have proved its identity.

Okay, So I'm gonna just look in my left hand side and try to get the same as my right hand side. I have a Cosenza traction formula. So applying that formula, this will simplify his co sign of first angle times co sign of the second angle. Opposite operation Sort of subtraction addition. Sign of the first angle time. Sign of the second angle. Okay. Cosign avec stays a consonant piper to I know that that's gonna be Cirrus. This will cancel sign of x days and then sign a piper to just once I multiplying set of X by one, which is just sign of X. So my left hand side is seems my radiant side you.


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