We know that tan = opp / adj
arctan x = theta
Taking the tan of each side
x = tan theta = opp / adj
x/1 = tan theta
The hypotenuse :
We can find this using the pythagorean theorem
a^2 + b^2 = c^2
1^2 + x^2 = hyp ^2
The hypotenuse = sqrt( 1+x^2)
Now we can find the cos of the angle
cos theta = adj side/ hypotenuse
cos theta = 1/ sqrt(1+x^2)
Rationalizing the denominator
cos theta = sqrt(1+x^2)/ (1+x^2)
cos ( tan^-1 x) = sqrt(1+x^2)/ (1+x^2)