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Math Help - ACT Question - cos and tan

  1. #1
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    ACT Question - cos and tan

    Hello, I am preparing for the ACT and stumbled upon a question that I don't understand.

    It says, "If tan x and cot x are defined, what is the value of (tan2x)(cot2x)?" (the 2 representing a square)

    The possible choices are:
    a) 0
    b) 1
    c) sin2 x -1 / cos2 x
    d) tan2x + sec2x
    e) (sec2x + 1)(csc2x + 1)


    Of course, I don't need the answer; I need an explanation. Any help is appreciated!
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  2. #2
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    tan(x) * cot(x) = 1 if both are defined - which they are.

    Since tan(x) and cot(x) ALWAYS have the same sign.

    \tan^{2}(x) \cdot \cot^{2}(x) = \left[\tan(x) \cdot \cot(x) \right]^{2}

    Where does that leave us?
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  3. #3
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    Thank you for your reply! So the answer is 1.


    I've been out of high school for 4 years, and I failed trig, out of laziness sadly. I barely understand trig.

    1) I have no idea why 1 is the defined. Is it a math standard to use 1 when they are specified as defined?

    2) What do you mean when you refer to "sign"?

    Many thanks!
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  4. #4
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    Quote Originally Posted by calebp View Post
    Thank you for your reply! So the answer is 1.


    I've been out of high school for 4 years, and I failed trig, out of laziness sadly. I barely understand trig.

    1) I have no idea why 1 is the defined. Is it a math standard to use 1 when they are specified as defined?

    2) What do you mean when you refer to "sign"?

    Many thanks!
    1) The tangent and cotangent functions are not defined at certain angles. The tangent function can be written as tan(x)=\frac{sin(x)}{cos(x)}. When cos(x)=0, the denominator of the tangent function is obviously zero, so tangent isn't defined when cosine is zero (the denominator of a fraction can NEVER be zero). Cosine is zero at 90 degrees and 270 degrees, so tangent isn't defined for these angles.

    Cotangent is just the inverse of tangent, so it isn't defined when sine is equal to zero.



    2) He means positive or negative. When tangent is positive, cotangent is also positive and vice-versa.
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  5. #5
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    it can be written in this way, y = (tan2x)(cot2x).

    as long as the value of x not equal to 0, pi/2, pi, or 3pi/2, the function is defined.

    y = (tan^2 x)(cot^2 x) = (tan^2 x)(1/tan^2 x) = 1
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