Which value of a will make this true?
-2w = -arctan( w / a) ?
$\displaystyle \begin{align*}-2w &=-\tan^{-1}{\left( w \over a \right)} \\2w &=\tan^{-1}{\left( w \over a \right)}\\ \tan(2w) &= \frac{w}{a} \\ a &=w \cdot \cot(2w) \end{align*}$
The equation is true for $\displaystyle a=w\cdot \cot{(2w)}$.