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Math Help - Prove: (cosxcotx)/1-sinx - 1 = csc x

  1. #1
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    Prove: (cosxcotx)/1-sinx - 1 = csc x

    As the title says, Prove: (cosxcotx)/1-sinx - 1 = csc x

    Any help is appreciated! Thanks.
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  2. #2
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    Hello, SMA777!

    This is a strange one . . .


    Prove: . \frac{\cos x\cot x}{1-\sin x} - 1 \:= \:\csc x

    On the left side, multiply the fraction by \frac{1+\sin x}{1+\sin x}

    . \frac{\cos x\!\cdot\!\dfrac{\cos x}{\sin x}}{1-\sin x}\cdot{\color{blue}\frac{1+\sin x}{1+\sin x}} -1 \;=\;\frac{\dfrac{\cos^2\!x}{\sin x}(1 + \sin x)}{1 - \sin^2\!x}-1 \;=\;\frac{\dfrac{\cos^2\!x}{\sin x}(1 + \sin x)}{\cos^2x} -1


    . . = \;\frac{1 + \sin x}{\sin x} - 1 \;=\;\frac{1+\sin x - \sin x}{\sin x} \;=\; \frac{1}{\sin x} \;=\;\csc x

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  3. #3
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    Ah, thank you!
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