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Math Help - Identity and Curve

  1. #1
    danco
    Guest

    Identity and Curve

    Arghhh I'm really stuck on these 2 problems. I never seem to be able to get them out. Any help would be greatsly appreciated.


    Prove the identity:

    cosx/1-tanx + sinx/1-cotx = sinx + cosx


    and also:

    Eliminate x from the equations below:

    y = 3 - 2sinx
    x = 4 + 5cotx
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  2. #2
    Super Member

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    Hello, danco!

    Prove the identity: .cos x/(1 - tan x) + sin x/(1 - cot x) .= .sin x + cos x
    . . . . . . . . . . cos x . . . . .sin x
    We have: . ----------- + ------------
    . . . . . . . . . . .sin x . . . . . cos x
    . . . . . . . . 1 - ------ . . 1 - -------
    . . . . . . . . . . .cos x . . . . .sin x


    Multiply top and bottom of the first fraction by cos x.
    Multiply top and bottom of the second fraction by sin x.

    . . . . . . . . . . . . . cos▓x . . . . . . .sin▓x . . . . . . . . .cos▓x . . . . . . sin▓x
    And we have: . --------------- + -------------- . = . --------------- - ---------------
    . . . . . . . . . . . cos x - sin x . .sin x - cos x . . . . cos x - sin x . .cosx - sin x


    . . . . . . cos▓x - sin▓x . . . . (cos x - sin x)(cos x + sin x)
    . . . = . ---------------- . = . ---------------------------------- . = . cos x + sin x
    . . . . . . cos x - sin x . . . . . . . . . .cos x - sin x



    I suspect there are typos in this one.

    Eliminate θ from the equations below:

    . . y .= .3 - 2Ěsinθ
    . . x .= .4 + 5Ěcotθ . ← I bet this is cosine
    From the first equation: .y .= .3 - 2Ěsinθ . . sinθ .= .(3 - y)/2 .[1]

    From the second equation: .x .= .4 + 5Ěcosθ . . cosθ .= .(x - 4)/5 .[2]


    Square [1]: .sin▓θ .= .(3 - y)▓/4

    Square [2]: .cos▓θ .= .(x - 4)▓/25


    Add: .(3 - y)▓/4 + (x - 4)▓/25 .= .sin▓θ + cos▓θ .= .1


    . . . . . . . . . . . . . . . . . . (x - 4)▓ . (y - 3)▓
    And we have an ellipse: . --------- + -------- .= .1
    . . . . . . . . . . . . . . . . . . . .25 . . . . . 4

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