Question

cos(arctan4/5)

Answer 1

draw a triangle
label the "opposite side" 5 and the "adjacent side" 5 like this

Answer 2

i meant "opposite side 4" sorry

Answer 3

all that is missing is the hypotenuse, which you find via pythagoras is
\[\sqrt{4^2+5^2}=\sqrt{16+25}=\sqrt{41}\]

Answer 4

now you can find the cosine easily, since it is "adjacent over hypotenuse"