Answer :
Please find the attachment.
We have been given that △EFG has a right angle at F, EG=6, and FG=4. We are asked to find trigonometric ratio for given angles of the triangle.
First of all, we will draw a right triangle using our given information.
Now we will find length of side EF using Pythagoras theorem.
[tex]EF^2=EG^2-FG^2[/tex]
[tex]EF^2=6^2-4^2[/tex]
[tex]EF^2=36-16[/tex]
[tex]EF^2=20[/tex]
[tex]EF=\sqrt{20}=2\sqrt{5}[/tex]
We know that cosecant relates hypotenuse with opposite side of right triangle.
[tex]\csc=\frac{\text{Hypotenuse}}{\text{Opposite}}[/tex]
We can see that opposite side to angle E is FG and hypotenuse is EG.
[tex]\csc(E)=\frac{6}{4}[/tex]
[tex]\csc(E)=\frac{3}{2}[/tex]
We know that cosine relates adjacent side with hypotenuse.
[tex]\cos=\frac{\text{Adjacent}}{\text{Hypotenuse}}[/tex]
We can see that adjacent side to angle G is FG and hypotenuse is EG.
[tex]\cos(G)=\frac{4}{6}[/tex]
[tex]\cos(G)=\frac{2}{3}[/tex]
We know that cotangent relates adjacent side with opposite side of right triangle.
[tex]\cot=\frac{\text{Adjacent}}{\text{Opposite}}[/tex]
We can see that adjacent side to angle G is FG and opposite side is EG.
[tex]\cot(G)=\frac{4}{2\sqrt{5}}[/tex]
[tex]\cot(G)=\frac{2}{\sqrt{5}}[/tex]
