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Math Help - proofs - help

  1. #1
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    proofs - help

    Cos (x-y)
    ------------ =
    Cos (x+y)

    1 + tanx tany
    --------------
    1 - tanx tany

    and

    Sin 3x=3sinx-4sin^3x


    Any help would be greatly appreciated
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by mortifiedpenguin View Post
    Cos (x-y)
    ------------ =
    Cos (x+y)

    1 + tanx tany
    --------------
    1 - tanx tany
    Spoiler:
    \frac{1+\tan x\tan y}{1-\tan x\tan y} = \frac{1+\tan x\tan y}{1-\tan x\tan y}\cdot\frac{\cos x\cos y}{\cos x\cos y}=\ldots


    and

    Sin 3x=3sinx-4sin^3x


    Any help would be greatly appreciated
    Spoiler:
    \begin{aligned}\sin(3x)&=\sin(2x)\cos x +\cos(2x)\sin x\\&=2\sin x\cos^2x+(1-2\sin^2x)\sin x\\&=\ldots\end{aligned}

    Continue applying identities to get everything in terms of sine and its powers. Then simplify to get the result!


    I hope these hints help!
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  3. #3
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    I would start with

    \frac{1+\tan(x)\tan(y)}{1-\tan(x)\tan(y)}

    and transform them into something to which you can apply the angle sum identities for cosine. (i.e. cos(x+y) = cos(x)cos(y)-sin(x)sin(y) and
    cos(x-y)=cos(x)cos(y)+sin(x)sin(y) )
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  4. #4
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    Hello, mortifiedpenguin!

    \frac{\cos(x-y)}{\cos(x+y)} \:=\:\frac{1 + \tan x\tan y}{1 - \tan x\tan y}

    We have: . \frac{\cos(x-y)}{\cos(x+y)} \;=\;\frac{\cos x\cos y + \sin x\sin y}{\cos x\cos y - \sin x\sin y}


    Divide numerator and denominator by \cos x\cos y

    . .  \dfrac{\dfrac{\cos x\cos y}{\cos x\cos y} + \dfrac{\sin x\sin y}{\cos x\cos y}} {\dfrac{\cos x\cos y}{\cos x\cos y} - \dfrac{\sin x\sin y}{\cos x\cos y}} \;=\;\frac{1 + \tan x\tan y}{1 - \tan x\tan y}

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  5. #5
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    sin 3x=3sinx - 4sin^3x

    sin 3x = sin( 2x +x )
    = sin2xcosx + sinxcos2x
    = (2sinxcosx)cosx + (cos^2x - sin^2x)sinx
    = 2sinxcosx^2 + cos^2xsinx - sin^3x
    = ...
    Last edited by Chris L T521; May 28th 2011 at 04:33 PM. Reason: edited out last part of full solution.
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