Answer
$\cot(\tan^{-1}(2x)) = \frac{1}{2x}$
Work Step by Step
We are trying to find $\cot(\tan^{-1}(2x))$. We can construct a right triangle that can help us find the value of the expression. Since the expression has a $\tan^{-1}(2x)$ term, we will construct a right triangle with legs of lengths $1$ and $2x$ and a hypotenuse of $\sqrt{1+4x^2}$. Let $\theta$ be the angle between the leg with length $1$ and hypotenuse. We see that $\tan(\theta) = 2x$ and solving for $\theta$, we conveniently get $\theta = \tan^{-1}(2x)$, which is part of our expression. We can substitute this into the original expression to get $\cot(\tan^{-1}(2x)) = \cot(\theta) = \frac{1}{2x}$.
