What are the asymptote(s) and hole(s), if any, of $f (x) = tan x \cdot csc x$ $f(x) = tanx*cscx$ ?

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Answer 1 · 2018-08-05T06:43:44

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

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]}

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