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Thread: prove the identity is true....

  1. #1
    Senior Member
    Aug 2009

    prove the identity is true....

    arctan x + arctan(1/x) = pi/2

    the way the book did it was weird, it substituted or allowed y to be equal arctan x + arctan (1/x) after manipulating the numbers it solved that y is undefined and therefore the function is true?? i know there is an easier and more logical way to do this problem. arctan x = y so tan y = x and if arctan (1/x) = y then tan y = 1/x, i jus dont see how adding those two parts will give me pi/2? any hints, pointers, advice? thanks in advance...
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  2. #2
    Senior Member
    Jul 2009
    Hi slapmaxwell1

    Not sure but maybe this can be right...

    $\displaystyle \tan(tan^{-1}x+tan^{-1}(\frac{1}{x}))$

    $\displaystyle =\frac{tan(tan^{-1}x)+tan(tan^{-1}(\frac{1}{x}))}{1-tan(tan^{-1}x)*tan(tan^{-1}(\frac{1}{x}))}$

    $\displaystyle =\frac{x+\frac{1}{x}}{1-x*\frac{1}{x}}$

    $\displaystyle =\infty$

    $\displaystyle =tan(\frac{\pi}{2})$

    So, arctan x + arctan(1/x) = pi/2
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