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Math Help - Function simplification

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    Function simplification

    I've tried to simplify in many ways with using identities, but I'm stuck.

    Simplify:

    \frac{\cos(x)}{\csc(x)-\sin(x)}

    Any ideas? Thanks!
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  2. #2
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    Multiply top & bottom by \sin x

    Apply one useful identity.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Failbait View Post
    I've tried to simplify in many ways with using identities, but I'm stuck.

    Simplify:

    \frac{\cos(x)}{\csc(x)-\sin(x)}

    Any ideas? Thanks!
    note that \csc x = \frac 1{\sin x}

    thus you have \frac {\cos x}{\frac 1{\sin x} - \sin x}

    now multiply the top and bottom through by \sin x and continue
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  4. #4
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    Quote Originally Posted by Failbait View Post
    I've tried to simplify in many ways with using identities, but I'm stuck.

    Simplify:

    \frac{\cos(x)}{\csc(x)-\sin(x)}

    Any ideas? Thanks!
    \frac{cos(x)}{csc(x)-sin(x)}

    = \frac{cos(x)}{\frac{1}{sin(x)} - sin(x)}

    = \frac{cos(x)}{\frac{1}{sin(x)} - sin(x)} \cdot \frac{sin(x)}{sin(x)}

    = \frac{sin(x)cos(x)}{1 - sin^2(x)}

    = \frac{sin(x)cos(x)}{cos^2(x)}

    = \frac{sin(x)}{cos(x)}

    = tan(x)

    (But note that we require that sin(x) \neq 0 from the original expression.)

    -Dan
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