# How do you simplify (cot(theta))/ (csc(theta) - sin(theta))?

Sep 30, 2016

$= \frac{\cos \frac{\theta}{\sin} \theta}{\frac{1}{\sin} \theta - \sin \theta}$

$= \frac{\cos \frac{\theta}{\sin} \theta}{\frac{1}{\sin} \theta - {\sin}^{2} \frac{\theta}{\sin} \theta}$

=(costheta/sintheta)/((1 - sin^2theta)/sintheta

$= \frac{\cos \frac{\theta}{\sin} \theta}{{\cos}^{2} \frac{\theta}{\sin} \theta}$

$= \cos \frac{\theta}{\sin} \theta \times \sin \frac{\theta}{\cos} ^ 2 \theta$

$= \frac{1}{\cos} \theta$

$= \sec \theta$

Hopefully this helps!

Sep 30, 2016

$\sec \theta$

#### Explanation:

Since $\cot \theta = \cos \frac{\theta}{\sin} \theta \mathmr{and} \csc \theta = \frac{1}{\sin} \theta$, the expression becomes:

$\frac{\cos \frac{\theta}{\sin} \theta}{\frac{1}{\sin} \theta - \sin \theta}$

that's

$\frac{\cos \frac{\theta}{\sin} \theta}{\frac{1 - {\sin}^{2} \theta}{\sin} \theta}$;

then, since $1 - {\sin}^{2} \theta = {\cos}^{2} \theta$, the expression becomes:

$\frac{\cos \frac{\theta}{\cancel{\sin}} \theta}{{\cos}^{2} \frac{\theta}{\cancel{\sin}} \theta}$

$= \frac{1}{\cos} \theta = \sec \theta$