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Math Help - tan identity

  1. #1
    Newbie
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    tan identity

    if anyone could show me how to do this that would be awesome

    show that if: <br />
tan(2y)=\frac{1}{tan(x)}\

    then this is equivalent to the expression: y=\frac{\Pi}{4}-\frac{x}{2}

    thanks
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, biker.josh07!

    I have a "visual" solution.



    Show that if: \tan(2y)\:=\:\frac{1}{\tan x}\

    then this is equivalent to the expression: y\:=\:\frac{\pi}{4}-\frac{x}{2}

    Since \tan2y \:=\:\frac{1}{\tan x}, then two angles are complementary.



    Consider this right triangle:


    Code:
                          *
                        * |
                      * x |
                    *     |
                  *       | a
                *         |
              *           |
            * 2y          |
          * - - - - - - - *
                  b

    \text{Since }\tan 2y \,=\,\frac{a}{b}\:\text{ and }\:\tan x \,=\,\frac{b}{a},\:\text{ then: }\:\tan2y \,=\,\frac{1}{\tan x}

    . . x and 2y are in the same right triangle.


    Hence: . x + 2y \:=\:\frac{\pi}{2} \quad\Rightarrow\quad 2y \:=\:\frac{\pi}{2} - x


    Therefore: . y \;=\;\frac{\pi}{4} - \frac{x}{2}

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