Function simplification

**Failbait** · October 14th 2007, 03:00 PM

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!

**Krizalid** · October 14th 2007, 03:11 PM

Multiply top & bottom by $\sin x$

Apply one useful identity.

**Jhevon** · October 14th 2007, 03:13 PM

Originally Posted by Failbait

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

**topsquark** · October 14th 2007, 03:14 PM

Originally Posted by Failbait

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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