1. Function 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!

2. Multiply top & bottom by $\displaystyle \sin x$

Apply one useful identity.

3. Originally Posted by Failbait
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!
note that $\displaystyle \csc x = \frac 1{\sin x}$

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

now multiply the top and bottom through by $\displaystyle \sin x$ and continue

4. Originally Posted by Failbait
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