Recall that the cosine ratio is determined by adjacent side, divided by the hypotenuse of a right triangle.
Given that cosine is 4/5, the opposite side is
[tex]\begin{gathered} a^2+b^2=c^2 \\ a^2+(4)^2=(5)^2 \\ a^2+16=25 \\ a^2=25-16 \\ a^2=9 \\ \sqrt{a^2}=\sqrt{9} \\ a=3 \end{gathered}[/tex]Now that we have solved for the opposite side, recall that cosecant is determined by the equation
[tex]\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}[/tex]Substitute
hypotenuse = 5
opposite = 3
and we get
[tex]\csc\theta=\frac{5}{3}\text{ \lparen final answer\rparen}[/tex]