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Math Help - Proving Trig Identities

  1. #1
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    Proving Trig Identities

    I'm having troubles proving some trig identities. Any help is appreciated.

    Question 1:
    1 - sin2x / cos2x = 1 - tanx / 1 + tanx

    Ignore question 2, I figured it out.

    Question 2:
    1 + sin2x / cosx + sinx = cos2x / cosx - sinx
    Last edited by peekaboos; November 12th 2008 at 06:17 PM.
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  2. #2
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    Hello, peekaboos!

    1) Prove: . \frac{1-\sin2x}{\cos2x} \;=\;\frac{1-\tan x}{1 + \tan x}

    The right side is: . \frac{1-\tan x}{1 + \tan x} \;=\;\frac{1 - \frac{\sin x}{\cos x}}{1 + \frac{\sin x}{\cos x}}


    Multiply by \frac{\cos x}{\cos x}\!:\quad \frac{{\color{blue}\cos x}\left(1 - \frac{\sin x}{\cos x}\right)} {{\color{blue}\cos x}\left(1 + \frac{\sin x}{\cos x}\right)} \;=\;\frac{\cos x - \sin x}{\cos x + \sin x}


    Multiply by \frac{\cos x - \sin x}{\cos x - \sin x}\!: \quad \frac{\cos x - \sin x}{\cos x + \sin x}\cdot{\color{blue}\frac{\cos x - \sin x}{\cos x - \sin x}} \;=\;\frac{(\cos x - \sin x)^2}{\cos^2\!x - \sin^2\!x}

    . . = \;\frac{\overbrace{\cos^2\!x + \sin^2\!x}^{\text{This is 1}} - \overbrace{2\sin x\cos x}^{\text{This is }\sin2x}}{\underbrace{\cos^2\!x - \sin^2\!x}_{\text{This is }\cos2x}} \;=\;\frac{1-\sin2x}{\cos2x}

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