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Math Help - Prove identity problem

  1. #1
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    Prove identity problem

    I have to prove that:

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

    How would I go about doing this? I've tried several things, to no avail. Any help?

    Thanks,
    Jason
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  2. #2
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    Lexington, MA (USA)
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    Hello, Jason!

    Prove: . \frac{1-\cos x}{\sin x} + \frac{\sin x}{1-\cos x}\; = \;2\csc x

    I've tried several things, to no avail.
    I don't suppose you tried adding those two fractions . . .


    \frac{1-\cos x}{\sin x} + \frac{\sin x}{1-\cos x} \;=\;{\color{blue}\frac{1-\cos x}{1-\cos x}}\cdot\frac{1-\cos x}{\sin x} + {\color{blue}\frac{\sin x}{\sin x}}\cdot\frac{\sin x}{1-\cos x}

    . . = \;\frac{(1-\cos x)^2 + \sin^2x}{\sin x(1-\cos x)}\;=\;\frac{1 - 2\cos x + \overbrace{\cos^2x + \sin^2x}^{\text{This is 1}}}{\sin x(1 - \cos x)}

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

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  3. #3
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    Hehe, I didn't think it would be that simple :P

    Thanks much!
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