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Math Help - Trigonometry: Using Double Angle Formula for Tangent to Prove an Equality

  1. #1
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    Trigonometry: Using Double Angle Formula for Tangent to Prove an Equality

    1. The problem statement, all variables and given/known data
    Show that tan(4arctan(1/5)) is equivalent to 120/119


    2. Relevant equations
    Double Angle Formula for tan:
    tan(2A) =\frac{2tanA}{1-\tan^2A}


    3. The attempt at a solution
    From my understanding you have to apply the double angle formula twice.
    Set x = arctan(1/5) and plug that into the double angle formula to solve for tan(2arctan1/5) (which is basically 2x). Now I set 2x = y, and plug y into the double angle formula to solve for 2y, which is 4arctan1/5.

    sub x = arctan(1/5) into the double angle formula:
    tan(2A) =\frac{2tanA}{1-\tan^2A}
    tan(2arctan(1/5)) =\frac{2tan(arctan1/5)}{1-\tan^2(arctan1/5)}
    Apply tan(arctanx) = x into the formula, resulting with:
    tan(2arctan(1/5)) =\frac{2(1/5)}{1-1/5arctan1/5}
    Simplify the numerator:
    tan(2arctan(1/5)) =\frac{2/5}{1-1/5arctan1/5}

    At this point I'm stuck, did I mess up in simplifying the denominator because I would like to be able to simplify it further so I don't have to use the final result above when I plug it back into a double angle formula once more. If that is correct, how would I work solve for
    tan(2(\frac{2/5}{1-1/5arctan1/5})) =\frac{2tan(\frac{2/5}{1-1/5arctan1/5})}{1-\tan^2(\frac{2/5}{1-1/5arctan1/5})}?

    Thanks for any assistance.
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  2. #2
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    Hi Ali812, welcome to MHF.
    2arctan(1/5) = arctan[2/5/(1-1/25) = arctan(10/24) = arctan(5/12)
    Similarly
    2arctan(5/12) = ...........?
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  3. #3
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    Thumbs up

    Quote Originally Posted by sa-ri-ga-ma View Post
    Hi Ali812, welcome to MHF.
    2arctan(1/5) = arctan[2/5/(1-1/25) = arctan(10/24) = arctan(5/12)
    Similarly
    2arctan(5/12) = ...........?
    2arctan(5/12) = arctan[2(5/12)/(1-25/144)] = arctan[(10/12)/(119/144)] = arctan[(120/144)/(119/144)] = arctan[120/119]

    And since I'm plugging these into a double formula for tangent, the tan and arctan cancel out and I'm left with 120/119! Wow thank you very much!
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