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!
$\displaystyle \frac{cos(x)}{csc(x)-sin(x)}$
$\displaystyle = \frac{cos(x)}{\frac{1}{sin(x)} - sin(x)}$
$\displaystyle = \frac{cos(x)}{\frac{1}{sin(x)} - sin(x)} \cdot \frac{sin(x)}{sin(x)}$
$\displaystyle = \frac{sin(x)cos(x)}{1 - sin^2(x)}$
$\displaystyle = \frac{sin(x)cos(x)}{cos^2(x)}$
$\displaystyle = \frac{sin(x)}{cos(x)}$
$\displaystyle = tan(x)$
(But note that we require that $\displaystyle sin(x) \neq 0$ from the original expression.)
-Dan