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Math Help - prove tanx =

  1. #1
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    prove tanx =

    hi

    I'm trying to prove: tanx = x + (1/3)x^3 + (2/15)x^5 +...

    using sinx = x - (1/3!)x^3 + (1/5!)x^5 -...

    and cosx = 1 - (1/2!)x^2 + (1/4!)x^4 -...

    my working is as follows:

    tanx=sinx/cosx = (sinx)(cosx)^-1

    I then take

    (cosx)^-1 = [1 - (1/2!)x^2 + (1/4!)x^4 -...]^-1 ......(i)

    and try using the binomial theorem

    (1 + x)^-1 = 1 -x + x^2 - x^3 +... .......(ii)

    but when i try to substitute (i) into (ii) i come unstuck!!


    help please.
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    South Coast of England
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    Hello 1cookie
    Quote Originally Posted by 1cookie View Post
    hi

    I'm trying to prove: tanx = x + (1/3)x^3 + (2/15)x^5 +...

    using sinx = x - (1/3!)x^3 + (1/5!)x^5 -...

    and cosx = 1 - (1/2!)x^2 + (1/4!)x^4 -...

    my working is as follows:

    tanx=sinx/cosx = (sinx)(cosx)^-1

    I then take

    (cosx)^-1 = [1 - (1/2!)x^2 + (1/4!)x^4 -...]^-1 ......(i)

    and try using the binomial theorem

    (1 + x)^-1 = 1 -x + x^2 - x^3 +... .......(ii)

    but when i try to substitute (i) into (ii) i come unstuck!!


    help please.
    Instead of your equation (ii), I'll use one which is basically the same: (1-x)^{-1}= 1 + x + x^2 + ... and I'm assuming that we can ignore all terms in x of higher power than x^5.

    So \cos x = 1 - \frac{x^2}{2}+\frac{x^4}{24}= 1 - \Big(\frac{x^2}{2}-\frac{x^4}{24}\Big)

    \Rightarrow (\cos x)^{-1} = 1 + \Big(\frac{x^2}{2}-\frac{x^4}{24}\Big) +\Big(\frac{x^2}{2}-\frac{x^4}{24}\Big)^2+...

    = 1 +\frac{x^2}{2}-\frac{x^4}{24} + \Big(\frac{x^2}{2}\Big)^2+...

    = 1 +\frac{x^2}{2}-\frac{x^4}{24} + \frac{x^4}{4}

    = 1 + \frac{x^2}{2}+\frac{5x^4}{24}

    So \tan x =(\sin x)(\cos x)^{-1}= \Big(x - \frac{x^3}{6}+\frac{x^5}{120}\Big) \Big(1 + \frac{x^2}{2}+\frac{5x^4}{24}\Big)

    = x + x^3\Big(-\frac{1}{6}+\frac{1}{2}\Big)+ x^5\Big(\frac{5}{24}-\frac{1}{12}+\frac{1}{120}+...\Big)

    = x +\tfrac13x^3 +\tfrac{2}{15}x^5

    Grandad
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  3. #3
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    Oct 2007
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    ...ah i see

    many thanks Grandad, my algebra lets me down from time to time.

    thanks once again for all your help!!

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