Let's assume by arctan xarctanx we mean the multivalued inverse, all the angles whose tangent is xx.
The double angle formula for sine is
sin (2 theta) = 2 sin theta cos theta sin(2θ)=2sinθcosθ
sin(2 arctan x) = 2 sin(arctan x) cos(arctan x)sin(2arctanx)=2sin(arctanx)cos(arctanx)
We can think of arctan (x/1) arctan(x1) as a right triangle whose opposite is xx and adjacent is 11, so the hypotenuse is sqrt{1+x^2}.√1+x2.
The sign of sine and of cosine are each ambiguous when we know the tangent. But the sign of the product sine and cosine and the sign of the tangent (which is the quotient of sine and cosine) are the same.
sin arctan x = text{opp}/text{hyp} = pm x/sqrt{1+x^2}sinarctanx=opphyp=±x√1+x2
cos arctan x = text{opp}/text{hyp} = pm 1/sqrt{1+x^2}cosarctanx=opphyp=±1√1+x2
sin(2 arctan x) = 2 sin(arctan x) cos(arctan x) = {2x}/{1+x^2}sin(2arctanx)=2sin(arctanx)cos(arctanx)=2x1+x2
Despite the ambiguity of the sign of the factors, the sign of the sine of the double angle is not ambiguous.
