From the trigonometric identity #sin^2(theta) + cos^2(theta) = 1#, divide both sides by #cos^2(theta)#

#sin^2(theta)/cos^2(theta) + cos^2(theta)/cos^2(theta) = 1/cos^2(theta)#

#tan^2(theta) + 1 = sec^2(theta)#

Substitute #theta# for #arctan(2)#

#tan^2(arctan(2)) + 1 = sec^2(arctan(2))#

Since #tan(arctan(x)) = x# axiomatically, we have that

#sec^2(arctan(2)) = (2)^2+1#

#sec^2(arctan(2)) = 5#

Take the root

#sec(arctan(2)) = +-sqrt(5)#

To pick the sign look at the range of the arctangent. During this range #(-pi/2,pi/2)# the cosine is always positive, and therefore so is the secant

#sec(arctan(2)) = sqrt(5)#