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

Sep 5, 2016

$\csc \frac{- x}{\cot} \left(- x\right) = - \csc \frac{x}{-} \cot x = \frac{\csc x}{\cot x} = \frac{\frac{1}{\sin} x}{\cos \frac{x}{\sin} x}$

$= \frac{1}{\cos} x = \sec x$.

Sep 5, 2016

$\sec \left(x\right)$

#### Explanation:

We have: $\frac{\csc \left(- x\right)}{\cot \left(- x\right)}$

Let's apply the fact that $\csc \left(x\right)$ and $\cot \left(x\right)$ are odd functions:

$= \frac{- \csc \left(x\right)}{- \cot \left(x\right)}$

$= \frac{\csc \left(x\right)}{\cot \left(x\right)}$

Then, let's apply two standard trigonometric identities; $\csc \left(x\right) = \frac{1}{\sin \left(x\right)}$ and $\cot \left(x\right) = \frac{\cos \left(x\right)}{\sin \left(x\right)}$:

$= \frac{\frac{1}{\sin \left(x\right)}}{\frac{\cos \left(x\right)}{\sin \left(x\right)}}$

$= \frac{1}{\sin \left(x\right)} \cdot \frac{\sin \left(x\right)}{\cos \left(x\right)}$

$= \frac{1}{\cos \left(x\right)}$

Finally, let's apply another standard trigonometric identity; $\sec \left(x\right) = \frac{1}{\cos \left(x\right)}$:

$= \sec \left(x\right)$