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Verifying Identities.

  1. Jul 28, 2004 #1
    Hi all, I have a question proving an identity:

    [tex]sin(4x) = 4sin(x)cos^3(x)-4sin^3(x)cos(x)[/tex]

    I can't seem to figure it out. I know I should be using the known identities:

    [tex]sin(2x) = 2sin(x)cos(x)[/tex]

    and probably:

    [tex]cos(2x) = 1-2sin^2(x)[/tex]

    but I'm stuck. Please help!

    Thanks!
     
  2. jcsd
  3. Jul 29, 2004 #2

    shmoe

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    Science Advisor
    Homework Helper

    First hint, apply your double angle formula for sin to:

    [tex]sin(2(2x))[/tex]

    what do you get?

    Second hint, try factoring the right hand side to help see what identities you'll need:

    [tex]sin(4x) = 4sin(x)cos(x)(cos^2(x)-sin^2(x))[/tex]

    How does this compare to your answer above? Can you see where the factors come from?
     
  4. Jul 29, 2004 #3
    Mathematicians never memorize trig identities. Trig identities are proven using complex numbers. You can if you wish try to prove your identity with complex numbers, you may find it may be easier. If I can remember correctly cos(x)=(e^(ix) +e^(-ix))/2 and sin(x) = (e^(ix))-e^(-ix))/2i (someone correct me if I am wrong). From there all you have to do is algebra instead of working with identities to get the left side= to the right side.
     
  5. Jul 29, 2004 #4
    Rather one may also use D'Moivre,
    (cos4x+isin4x) = (cosx+isinx)^4
    expand RHS and compare imaginary parts ... as a bonus u also get cos4x :P
     
  6. Jul 29, 2004 #5
    let me try
    sin (4x) = 4 sin x cos *3 x - 4 sin *3 ( x ) cos ( x)

    let 2x = X
    sin 2X = 2 sin X cos X
    =2 sin 2x cos 2x
    =2 [( 2sinx cosx ) ( cos*2 x - sin *2 x ) ]
    =2 [ 2sinxcos*3 x - 2sin*3cos x ]
    = 4sinx cos*3 x - 4sin*3 x cos x
     
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