# How do you simplify costheta/(sin^2theta-1)?

$- \sec \theta$, $\theta \ne \frac{\pi}{2} \pm n 2 \pi$ where n is a natural number

#### Explanation:

We have:

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

Remember that there is an identity:

${\sin}^{2} \theta + {\cos}^{2} \theta = 1$

and so

${\cos}^{2} \theta = - {\sin}^{2} \theta + 1$ which can be written as:

${\cos}^{2} \theta = - \left({\sin}^{2} \theta - 1\right)$ and so

$- {\cos}^{2} \theta = \left({\sin}^{2} \theta - 1\right)$

so let's substitute that into our original statement:

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

$\cos \frac{\theta}{- {\cos}^{2} \theta}$

and now we can simplify:

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

Ok, so now what are the limitations? It's ${\sin}^{2} \theta \ne 1$ (that would make the denominator in our original question be 0) and so:

$\theta \ne \frac{\pi}{2} \pm n 2 \pi$ where n is a natural number