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Math Help - trig identity for sqrt(2)/2

  1. #16
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    Well, obviously it's going to simplify in such a way as to equal sqrt(2)/2. That's sort of the whole point of an identity isn't it.

    The challenge was to come up with a trig identity that equals this and that is valid for all angles without using the number two. A couple days ago you said such a thing didn't exist. I've demonstrated that it does exist. That seems to me to constitute something new.
    Last edited by rainer; December 13th 2009 at 05:34 AM. Reason: typo
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  2. #17
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    Ok. Why ask us?

    Go here:
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    Click on: Contribute to MathWorld

    Send them what you think is your discovery.
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  3. #18
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    The whole point of an identity is that it's not derived strictly from other identities. Take for example the pythagorean identity sin^2\theta + cos^2\theta = 1. As far as I'm concerned, it can not be derived from any algebraic derivation of another identity (which is exactly what your "identity" is...)

    Now,
    \frac{1}{1 + 2sin^2\theta + cos(2\theta)} \equiv \frac{1}{\sqrt{2}} \forall \theta, and I might as well come up with 50 other "identities", all equal \frac{1}{\sqrt{2}}, which are all based upon the pythagorean identity.

    If I saw \frac{1}{\sqrt{\cos^2\theta(\tan^2\theta+\sec^2\th  eta+1)}} somewhere, I would be able to simplify it to \frac{1}{\sqrt{2}}. But can you say that you will know how to show sin^2\theta + cos^2\theta = 1?

    Do you understand why identities are called as such?
    Last edited by Defunkt; December 13th 2009 at 10:13 AM.
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  4. #19
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    Defunkt,

    Before seeing your reply I was thinking this over and realized you are right. What I have come up with is a tautology, not an identity.

    It does still work as a puzzle, though. Another good puzzle would be to express the same thing with hyperbolic functions without using the number 2.

    As someone trying to understand math without a professor and without books, I must often rely on folly as my teacher. At the very least I hope my folly has provided some comic relief for everyone.
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  5. #20
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    Quote Originally Posted by rainer View Post
    At the very least I hope my folly has provided some comic relief for everyone.
    I see nothing "comically wrong" with your "buts, ifs..."; shows you have
    an inquisitive mind; also quite a good way to learn "by default"
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  6. #21
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    Quote Originally Posted by Defunkt View Post
    Take for example the pythagorean identity sin^2\theta + cos^2\theta = 1. As far as I'm concerned, it can not be derived from any algebraic derivation of another identity (which is exactly what your "identity" is...)
    Really?
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