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Math Help - Range

  1. #1
    Member kjchauhan's Avatar
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    Range

    Please help me to find the range of sec^{4}(x)+cosec^{4}(x).

    Thanks in advance.
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  2. #2
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    Re: Range

    Quote Originally Posted by kjchauhan View Post
    Please help me to find the range of sec^{4}(x)+cosec^{4}(x).
    What have you done?

    Have you at least graphed this expression?
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  3. #3
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    Re: Range

    Quote Originally Posted by kjchauhan View Post
    Please help me to find the range of sec^{4}(x)+cosec^{4}(x).

    Thanks in advance.
    \sec^4(x) = (\sec^2(x))^2 = (1+\tan^2(x))^2 = \tan^4(x) + 2\tan^2 + 1

    \csc^4(x) = (\csc^2(x))^2 = (1+\cot^2(x))^2 = \cot^4(x) + 2\cot^2 + 1


    \sec^4(x) + \csc^4(x) = \tan^4(x) + 2\tan^2 + 1 + \cot^4(x) + 2\cot^2 + 1  = \tan^4(x) + \cot^4(x) + 2(\tan^2(x)+\cot^2(x)) + 2

    The range of \tan(x) is (-\infty, \infty). Likewise, the range of \cot(x) is (-\infty, \infty).

    Since \tan^4(x) and \cot^4(x) are positive even powers, both have range [0,\infty).

    BELOW is where others may disagree with me:

    I contend that the range of \tan^4(x)+\cot^4(x) is (0,\infty) as opposed to [0,\infty), which the sum of the parts may intuitively suggest.
    The two ranges differ insofar (0,\infty) does not contain 0, whereas [0,\infty) does contain 0.
    I believe that the range of \tan^4(x)+\cot^4(x) should NOT include 0, i.e., it should be (0,\infty).

    Proof is achieved if we show \tan^4(x)+\cot^4(x)>0 on the entire domain (reals that are not multiples of \pi).

    Both terms in \tan^4(x)+\cot^4(x) are non-negative in the reals, so clearly the sum itself is non-negative.

    \tan^4(x)+\cot^4(x) cannot equal 0 because then either \tan^4(x)=-\cot^4(x) (which per the line above is not possible) OR \tan^4(x)=\cot^4(x)=0, which also can't happen because \cot^4(x)=\tfrac{1}{\tan^4(x)} and 0 cannot be a denominator.

    As such, \tan^4(x)+\cot^4(x)>0.

    And with this new information, \sec^4(x) + \csc^4(x) = \underbrace{\left[\tan^4(x) + \cot^4(x)\right]}_{\text{always positive!}} + 2\underbrace{(\tan^2(x)+\cot^2(x))}_{\text{can show positive similarly}} + 2 > 2.

    It at long last follows that the range of \sec^4(x) + \csc^4(x) is \left(2,\infty\right).

    If someone has an argument for a left bracket instead of my left open parenthesis, I'd love to hear it!

    -Andy
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    Re: Range

    Quote Originally Posted by abender View Post
    [TEX]\sec^4(x) = (\sec^2(x))^2 =
    I contend that the range of \tan^4(x)+\cot^4(x) is (0,\infty) as opposed to [0,\infty), which the sum of the parts may intuitively suggest.
    It at long last follows that the range of \sec^4(x) + \csc^4(x) is \left(2,\infty\right).

    If someone has an argument for a left bracket instead of my left open parenthesis, I'd love to hear it!
    Take a look at the graph of the function.
    Attached Thumbnails Attached Thumbnails Range-untitled.gif  
    Thanks from abender
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  5. #5
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    Re: Range

    [8,\infty), evidently. My way was a fun adventure; I wanted to do it using identities for some bizarre reason. These problems nearly always boil down to the rudimentary sines and cosines. I see the airtight solution now, the symmetry, no plus 2 times "something positive". In retrospect, why the hell did I accept 2 as my best lower bound? Oh well, my Ravens are in the Super Bowl! Usually never post if I'm uncertain.
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