Question 1180471


 {{{cos (t) = 3/5}}

 and the terminal point for {{{t}}} is in quadrant IV, 

find {{{cot( t) + csc (t)}}}

recall:

 {{{cos (t) = adj/hyp}}}=> if  {{{cos (t) = 3/5}}=>{{{ adj=3}}} and {{{hyp=5}}

then {{{opp=sqrt(5^2-3^2)}}}=>{{{opp=sqrt(25-9)}}}=>{{{opp=sqrt(16)}}}=>{{{opp=4}}} or {{{opp=-4}}} 

then {{{sin(t)=4/5}}} or {{{sin(t)=-4/5}}}

 since given that the terminal point for {{{t}}} is in quadrant IV, sine is negative, so we will use
{{{sin(t)=-4/5}}}

using identities {{{cot( t) =cos (t)/sin(t)}}} and {{{csc (t)=1/sin(t)}}}, we have

{{{cot( t) + csc (t)=cos (t)/sin(t) +1/sin(t)}}}...substitute values above

{{{cot( t) + csc (t)=(3/5)/ (-4/5)+1/(-4/5)}}}

{{{cot( t) + csc (t)=(3/5+1)/(-4/5)}}}

{{{cot( t) + csc (t)=(8/5)/(-4/5)}}}

{{{cot( t) + csc (t)=-2}}}