Feel the Power of $\cos\left(2\theta\right)$!

Jimmy started with $\theta = \pi$. Playing around with the following double-angle cosine formula

$\cos\left(2\theta\right) = 2\cos^2\left(\theta\right) - 1$

He made the following list of all cosine values in the following descending angle order $\begin{array}{rl} \cos(\pi) &= -1\\ \cos\left(\dfrac{\pi}{2}\right) &= 0\\ \cos\left(\dfrac{\pi}{4}\right) &= \dfrac{\sqrt{2}}{2}\\ \cos\left(\dfrac{\pi}{8}\right) &= \dfrac{\sqrt{2 + \sqrt{2}}}{2}\\ \cos\left(\dfrac{\pi}{16}\right) &= \dfrac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2}\\ \cos\left(\dfrac{\pi}{32}\right) &= \dfrac{\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}}}{2}\\ \vdots & \quad \vdots \end{array}$ Noticing the increasing number of $\sqrt \$'s and 2's, he then wrote down $\lim_{n \rightarrow \infty} \cos\left(\dfrac{\pi}{2^n}\right) = \dfrac{\sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}}{2} = \dfrac{2}{2} = 1$ If he were to follow these similar steps from $\theta = \dfrac{\pi}{k}$, where $k > 0$, can he also achieve the following limit to be 1? $\lim_{n \rightarrow \infty} \cos\left(\dfrac{\pi}{2^n \cdot k}\right) = 1$