# What are the asymptote(s) and hole(s), if any, of  f(x) = tanx*cscx?

Aug 5, 2018

There are no holes and the asymptote are $\left\{\begin{matrix}x = \frac{\pi}{2} + 2 k \pi \\ x = \frac{3}{2} \pi + 2 k \pi\end{matrix}\right.$ for $k \in \mathbb{Z}$

#### Explanation:

We need

$\tan x = \sin \frac{x}{\cos} x$

$\csc x = \frac{1}{\sin} x$

Therefore,

$f \left(x\right) = \tan x \cdot \csc x = \sin \frac{x}{\cos} x \cdot \frac{1}{\sin} x = \frac{1}{\cos} x = \sec x$

There are asymptotes when

$\cos x = 0$

That is

$\cos x = 0 , \implies \left\{\begin{matrix}x = \frac{\pi}{2} + 2 k \pi \\ x = \frac{3}{2} \pi + 2 k \pi\end{matrix}\right.$

Where $k \in \mathbb{Z}$

There are holes at the points where $\sin x = 0$ but $\sin x$ does not cut the graph of $\sec x$

graph{(y-secx)(y-sinx)=0 [-10, 10, -5, 5]}