Question

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

Answer 1

There are no holes and the asymptote are `{(x=pi/2+2kpi),(x=3/2pi+2kpi):}` for `k in ZZ`

Explanation:

We need

`tanx=sinx/cosx`

`cscx=1/sinx`

Therefore,

`f(x)=tanx*cscx=sinx/cosx*1/sinx=1/cosx=secx`

There are asymptotes when

`cosx=0`

That is

`cosx=0, => {(x=pi/2+2kpi),(x=3/2pi+2kpi):}`

Where `k in ZZ`

There are holes at the points where `sinx=0` but `sinx` does not cut the graph of `secx`

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