What are the asymptote(s) and hole(s), if any, of # 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]}