# Question #cf9c7

Sep 5, 2016

$\sec \left(x\right) \csc \left(x\right) - \sec \left(- x\right) \csc \left(- x\right) = 2 \sec \left(x\right) \csc \left(x\right)$

#### Explanation:

For this problem, we will make use of the facts that

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

(these are due to sine being an odd function and cosine being an even function, together with the definitions of the secant and cosecant functions)

$\sec \left(x\right) \csc \left(x\right) - \sec \left(- x\right) \csc \left(- x\right)$

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

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

$= 2 \sec \left(x\right) \csc \left(x\right)$