# convert n*Arctan[x] to ArcTan[x]

In the result of solving the equation or the integral often get something like k*ArcTan[x], I prefer the ArcTan[x] form.

For example, given list1, how do I get list2? FullSimplify doesn't work, my current approach is too limited, is there a more general way?

list1={2 ArcTan[1/2 (1+Sqrt[5])],2 ArcTan[Sqrt[5]-2],2 ArcTan[1/5 (-3+Sqrt[34])],3 ArcTan[Sqrt[1/6]],4 ArcTan[1/3]};

FullSimplify[list1]
FullSimplify[list1==list2]

list1 /. 2 ArcTan[x_] :> ArcTan[Simplify[(2 x)/(1 - x^2)]] /; -1 < x < 1


$$\left\{2 \tan ^{-1}\left(\frac{1}{2} \left(\sqrt{5}+1\right)\right),-2 \tan ^{-1}\left(2-\sqrt{5}\right),2 \tan ^{-1}\left(\frac{1}{5} \left(\sqrt{34}-3\right)\right),3 \cot ^{-1}\left(\sqrt{6}\right),4 \cot ^{-1}(3)\right\}$$

True

FunctionExpand + FullSimplify seems to work:
ArcTan @@ FullSimplify[FunctionExpand[{Cos[list1], Sin[list1]}]]

• +1. TrigExpand works too: ArcTan @@ TrigExpand@Through[{Cos, Sin}[list1]] // Simplify Commented Jun 3, 2022 at 17:46