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A Generalized trigonometric identity for Cos(x_1++x_n)?

  1. Jan 28, 2009 #1
    Hi Everyone,
    Do there exist any explicit formula for Cos(x_1+x_2+...+x_n) as a sum of products of Sin(x_i) & Cos(x_i)? Or we need to expand using Cos(A+B), Sin(A+B) again & again?
    If it exists then what is about Sin(x_1+x_2+...+x_n)?

    [It is understood that there will be 2^(n-1) number of terms in the sum each of which will be
    a product of n number of Sin(x_i) &/ Cos(x_i)]

    Regards
    Naturepaper
     
  2. jcsd
  3. Jan 28, 2009 #2

    Gib Z

    User Avatar
    Homework Helper

    http://en.wikipedia.org/wiki/Trigonometric_identity

    That link has some things that are related, but not exactly the same.

    I'm not sure if a formula can be found, but you can try it. Expand it for a couple of cases, try to find a formula and use induction.
     
  4. Jan 30, 2009 #3
    Thank you Gib Z for your reply. But the link you given doesn't answer the question. It was for infinite sum whereas I need for finite terms. Anyway I'm waiting for more reply...


    Cheers
    NaturePaper
     
  5. Jan 30, 2009 #4
  6. Feb 1, 2009 #5
    NaturePaper, the same page gives
    [tex]\sec(\theta_1 + \cdots + \theta_n) = \frac{\sec\theta_1 \cdots \sec\theta_n}{e_0 - e_2 + e_4 - \cdots}.[/tex]
    From that, it's obvious that
    [tex]\cos(\theta_1 + \cdots + \theta_n) = \frac{e_0 - e_2 + e_4 - \cdots}{\sec\theta_1 \cdots \sec\theta_n}.[/tex]
     
  7. Feb 1, 2009 #6
    @adriank,
    oh, very nice-I missed it. Thanks.

    Intuitively, Sin(x_1+x_2+...+x_n)=sinx_1.sinx_2....sinx_n(e_1-e_3+e_4-.......), where e_i are in terms of cotx_i. I'll check it.

    Actually I need a formula to convert the product of K number of sinx_i and (n-k) number of cosx_i into sum of Sin and cosine terms.

    If I get the expressions for Sin(x_1+x_2+...+x_n) and cos(x_1+x_2+...+x_n), after some manipulation I can get the required formula.

    Thanks & Regards
    NaturePaper
     
  8. Feb 4, 2009 #7
    Got it.

    Sin(x_1+x_2+...+x_n)=cosx_1cosx_2....cosx_n(e_1-e_3+e_4-...).

    By the way, I'll post a possible proof and detailed formula soon.

    cheers,
    NaturePaper
     
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