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Math Help - Need help simplifying an equation with arctan

  1. #1
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    Need help simplifying an equation with arctan

    How do I go from 1-(tan^2)(arctan x/y ) to 1 - (x^2)/(y^2)
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  2. #2
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    Re: Need help simplifying an equation with arctan

    Functions tan and arctan are inverse of each other. More precisely, tan is the left inverse of arctan everywhere, i.e., tan(arctan(x)) = x for all x and tan is the right inverse of arctan for x\in(-\pi/2,\pi/2), i.e., arctan(tan(x)) = x on this interval.
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    Re: Need help simplifying an equation with arctan

    I think what is confusing me is, why does the square on the tan^2(arctan....) etc get distributed to the x/y or is it more complicated than it being distributed. Also is arctan the same as 1/tan ? Thanks for the help, I am sorry if I confuse you but I want to know this well.
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    Re: Need help simplifying an equation with arctan

    Quote Originally Posted by Mouser View Post
    why does the square on the tan^2(arctan....) etc get distributed to the x/y
    The notation \tan^2(x) means (\tan(x))^2 by definition. So, \tan^2(\arctan(x/y))=(\tan(\arctan(x/y)))^2=(y/x)^2.

    Quote Originally Posted by Mouser View Post
    Also is arctan the same as 1/tan ?
    No, these are two very different things. See the graphs of the two functions here. For a given x, 1/tan(x) = cot(x) is the reciprocal of tan(x) whereas \tan^{-1}(y)=\arctan(y) is the unique x\in(\pi/2,\pi/2) such that tan(x) = y. Note that by analogy with \tan^2(x)=(\tan(x))^2, \tan^{-1}(y) should denote (\tan(y))^{-1}=1/\tan(y), but this is not so. Thus, \tan^n(x) denotes (\tan(x))^n for n > 0, but \tan^{-1}(x) usually denotes arctan(x), the inverse function of tan(x). This is somewhat confusing, but there is a tradition of denoting the inverse of a function f by f^{-1}.
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