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

Simplify:

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

Any ideas? Thanks!

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- Oct 14th 2007, 03:00 PMFailbaitFunction simplification
I've tried to simplify in many ways with using identities, but I'm stuck.

Simplify:

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

Any ideas? Thanks! - Oct 14th 2007, 03:11 PMKrizalid
Multiply top & bottom by $\displaystyle \sin x$

Apply one useful identity. - Oct 14th 2007, 03:13 PMJhevon
- Oct 14th 2007, 03:14 PMtopsquark
$\displaystyle \frac{cos(x)}{csc(x)-sin(x)}$

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

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

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

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

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

$\displaystyle = tan(x)$

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

-Dan