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Trigonometry: Proving an Identity: Other Examples

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About this Lesson

  • Type: Video Tutorial
  • Length: 6:16
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 67 MB
  • Posted: 07/02/2009

This lesson is part of the following series:

Trigonometry: Full Course (152 lessons, $148.50)
Trigonometry: Trigonometric Identities (23 lessons, $26.73)
Trigonometry: Proving Trigonometric Identities (2 lessons, $2.97)

Taught by Professor Edward Burger, this lesson comes from a comprehensive course, Trigonometry. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/trigonometry. The full course covers trigonometric functions, trigonometric identities, application of trig, complex numbers, polar coordinates, exponential functions, logarithmic functions, conic sections, and more.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

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Proving an Identitiy: Other Examples
Okay, so let's take a look at some more of these trig identities where you want to actually prove that the identity really holds. So let's take a look at this one. How about = tan x sin x. Let's see if we can prove that. Again, the way I think about this is sort of drawing a line in the sand. Maybe I'll draw the line a little bit skewed so I move them to right. The line is straight. And the idea is, I just work on this side, I just work on that side, and hopefully at the end of the day I should have two equal quantities.
So let's see how this might play out. Well, when you look at this, there are a variety of things you can do. But whenever I see a cosine squared and a 1 or -1, I'm immediately thinking about the Pythagorean identity, sin^2 + cos^2 = 1. In this case I have cos^2 x - 1, so what does that equal? Well, I've got the cos^2 x. If I bring this over, I have a -1, so what does that equal? I take this sin^2 and bring it over to this side. So I see that cos^2 x - 1 = -sin^2 x. So I have -sin^2 x. Let's put that in on the top. So this equals . Well, it's supposed to look like this. But what I do see--the minus signs, by the way, look very good. I hope you see that the minus signs make me very happy because that looks good. Well, sort of a pattern recognition thing out here. Let's write the sine out as just . And this , that can be combined to be what? Well, that's just tan x. And then I still have that sin x out in front. And viola! I see that this left-hand side is identical to the right-hand side. So this equals that, equals that, equals that, so these two things are equal. So I just proved that identity holds by just manipulating this guy using trig identities.
All right, let's try one last one. One last one, so it's sad for me because I love these things. All right, how about + tan and I want to see if that really equals sec csc. So sec csc, does that equal + tan? Well, how would you proceed here? Again, what I do is I draw a line in the sand and I'll start massaging the sides as necessary to see if I can make them look the same. Well, you could get a common denominator here, which would be tan. If I multiply top and bottom--the bottom here is invisible 1--top and bottom by , and then using trig identities using with tangent, that's fine. But I always use the same pathetic method. I convert everything to sines and cosines. Sometimes I may be losing ground here, by the way. I'm not saying this is always the best method, but it's the method that always works for me and I'm happy with it.
. Well, tan = , so when I flip it I get . And then I add tangent, which is . See, now I'm going to get a common denominator, which will require me to multiply top and bottom here by cosine. So when I do that, I'll see a . And then I multiply top and bottom here by sine. So I see . And now the bottoms are the same. I just add the tops and I see . But what's cos^2^^^^ + sin^2^^^^? Well, that equals 1. That's the Pythagorean identity. So cos^2^^^^ + sin^2^^^^ = 1, so in fact this is just . So that's this side.
Now, what about this side? What does that equal? Well, this side--let's see, what does cosecant equal? Cosecant equals, well, just . So this piece right here, this equals . And what's secant? Secant is . So I multiply. I get . And look! Once again we get to that moment in my life doing these problems where I just get really excited. and , these two things are equal. Well, since these are equal, that means that these are equal, these are equal, these are equal, these are equal, and these have to be equal. So we've actually proved the identity by working on this side, working on that side, and getting two quantities that are known to be equal.
Notice the key thing to remember here is that I can't use this as an identity. I can't add to both sides and divide both sides with stuff because I don't know it's an identity yet. I have to prove it's an identity by using just manipulations on one side and the other, and showing that it's actually the same. This is what we consider proving trigonometric identities, and I've got to tell you, I love them! I hope you love them, too. Actually, they're pretty fun. They're sort of like mind games. Try these mind games and have some fun on your own. See you later.

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