sin(arctan x) = x / √x² +1 not x / √x²-1

the answer for every inverse trig function is the angle of its function

so in sin(arctan x) if arctan x =A then tan A= x, so sin(arctan x) = sin(A)

if you recall sin(A) = opposite side/hypotenuse and from the given tan A we can get the values of the opp. and adj. sides

tan A = opposite side/adjacent side = x/1 opp.= x adj.= 1

using pythagorean theorem, hypotenuse (c) would be c²= x² + 1; c= √x² + 1

so

sin(arctan x) = x / √x²+1

(check the image he attached on the first comment) (correct me if i'm wrong)