Bridging Trig IdentitiesDate: 06/07/2002 at 11:08:59 From: Susannah Subject: trigonometric identities Hi...please help. I have been staring at this question for ages and I'm getting nowhere: Prove tan 2A x sec A = 2sin A x sec 2A My teacher told me that when proving these identities you have to choose one side which is what I have been doing. I started off by opening up the identity tan2A = 2tanA / 1 - tan*2 A and then that didn't get anywhere so I changed sides and opened up sec2A but that didn't help either. Thank you. Date: 06/07/2002 at 17:05:15 From: Doctor Peterson Subject: Re: trigonometric identities Hi, Susannah. Often when you are first working out a proof, it helps to work on both sides at once; then you can rewrite what you have done in a more proper way. This is sort of like constructing a bridge by building both ends toward the middle, but then driving across it in one direction. In this case, as in many, it helps a lot to write everything in terms of sine and cosine: tan(2A) * sec(A) =? sec(A) * sec(2A) sin(2A) 1 1 1 ------- * ------ =? ------ * ------- cos(2A) cos(A) cos(A) cos(2A) (Notice how I indicated that the "=" is not yet known to be true; that's my way to keep myself honest when I work this way!) Now you can apply a double-angle formula and do some canceling, and you will be able to make both sides look the same. Then you will have a real equation. Now how do we walk across the bridge and make it a proper proof? You will have written a sequence of equivalent equations of the form a =? b c =? d e = e (this one doesn't need the "?"!) As long as you have been working on each side separately, you can now say this: a b = = c d = = e = e That is, a = c = e = d = b and you have proved that a = b! Can you see now how we were working from both ends? We built the span on the left (a=c) and on the right (b=d) first, then the middle spans (c=e, d=e) and found we had connected the two shores. Then we just had to put it all in the right order and we were done. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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