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Prove Trig Identity

  1. Aug 3, 2004 #1
    How would you prove using the cancellation laws 2arccos(x) = arccos(2x² - 1). I am stumped. Any guidance is appreciated.
     
  2. jcsd
  3. Aug 3, 2004 #2

    uart

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    Start by taking cos of both sides to get :

    cos ( acos(x) + acos(x) ) = 2x^2 -1

    It's pretty easy from there, so I'll leave itfor you to have a go. :)
     
  4. Aug 3, 2004 #3
    What if I let θ = acos(x) which implies that x = cosθ

    Then the expression becomes...

    2θ = acos(2(cosθ )² - 1)
    2θ = acos(2cos²θ - 1)

    Then take cos of both sides and use the cancellation law...

    cos2θ = 2cos²θ - 1

    This is a well known trigonometric identity. So if we start from this and work backwards the identity in the question can easily be shown.
     
  5. Aug 3, 2004 #4
    Start with the double angle identity

    cos(2θ) = 2cos²θ -1

    Then take the square outside the brackets

    cos(2θ) = 2(cosθ )² -1

    Since cos(acos(x)) = x we may write it as

    cos(2θ) = cos(acos(2(cosθ )² -1))

    Now take inverse cos of both sides

    acos(cos(2θ)) = acos(cos(acos(2(cosθ )² -1)))

    Cancel out the acos(cos(x)) terms

    2θ = acos(2(cosθ )² -1)

    Let θ = acos(α) implies α = cosθ

    2acos(α ) = acos(2α² - 1)

    QED
     
  6. Aug 4, 2004 #5

    uart

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    I just used the usual cos(A+B) expansion on the LHS to give :

    cos^2( acos(x) ) - sin^2 (acos(x))

    and then added zero in the form of cos^2(.) + sin^2(.) - 1 to the LHS to give,

    2 cos^2( acos(x) ) - 1
     
  7. Aug 5, 2004 #6
    Another question.

    For what values of A is the trigonometric identity cos2A = 2cos²A - 1 valid? I thought it valid for all real numbers. But there must be a trick??

    Any hints would be appreciated. Thanks.
     
  8. Aug 5, 2004 #7
    It is valid for all A. It follows from the pythagorean identity and the identity cos (A+B) = cos A cos B - sin A sin B. That identity gives cos 2A = cos (A+A) = cos² A - sin² A. Adding on cos² A + sin² A - 1 (which is 0 by the pythagorean identity) to the right side gives the identity cos 2A = 2cos² A - 1
     
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