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Math Help - prove this identity

  1. #1
    Junior Member
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    prove this identity

    2sin(x)/[(cos(x)+sin(x)] = tan(2x) + 1 - sec(2x)

    First, I turn tan(2x) into [sin(2x) / cos(2x)] then I turn sec(2x) into 1/cos(2x)

    So, I end up with... [sin(2x) / cos(2x)] + 1 - (1/(cos2x))

    I multiple and get the GCD of cos(2x)..

    [sin(2x) + cos(2x) - 1] / cos(2x)

    Now things get kind of fuzzy.. I probably didn't do this the best way thus far but I can't really advance much further.. I need help, please.
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by zodiacbrave View Post
    2sin(x)/[(cos(x)+sin(x)] = tan(2x) + 1 - sec(2x)

    First, I turn tan(2x) into [sin(2x) / cos(2x)] then I turn sec(2x) into 1/cos(2x)

    So, I end up with... [sin(2x) / cos(2x)] + 1 - (1/(cos2x))

    I multiple and get the GCD of cos(2x)..

    [sin(2x) + cos(2x) - 1] / cos(2x)

    Now things get kind of fuzzy.. I probably didn't do this the best way thus far but I can't really advance much further.. I need help, please.
    Start with right side with the identy

    tan(2x)=\frac{2tanx}{1-tan^2x} and sec(2x)=\frac{1}{cos(2x)}=\frac{1+tan^2x}{1-tan^2x}

    Then the right side reduces to

    \frac{2tanx+2}{1-tan^2x}

    Chage tan to terms of sin and cos and you should be able to get there.
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  3. #3
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    Hello, zodiacbrave!

    Another approach . . .


    Prove: . \frac {2\sin x}{\cos x+\sin x} \:=\: \tan(2x) + 1 - \sec(2x)
    Multiply the left side by: . \frac{\cos x - \sin x}{\cos x - \sin x}

    \frac{2\sin x}{\cos x + \sin x}\cdot\frac{\cos x - \sin x}{\cos x - \sin x}

    . . =\;\frac{\overbrace{2\sin x\cos x}^{\sin(2x)} - \overbrace{2\sin^2\!x}^{1-\cos(2x)}}{\underbrace{\cos^2\!x - \sin^2\!x}_{\cos(2x)}}

    . . = \;\frac{\sin(2x) + \cos(2x) - 1}{\cos(2x)}

    . . =\;\frac{\sin(2x)}{\cos(2x)} + \frac{\cos x}{\cos x} - \frac{1}{\cos(2x)}

    . . =\;\tan(2x) + 1 - \sec(2x)

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  4. #4
    Senior Member pacman's Avatar
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    Soroban's LATEXing technique is superb and very instructive, thanks MHF expert
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