What is a fast way to evaluate $\cos(\pi/4 + 2\pi/3)$ and $\cos(\pi/4 + 4\pi/3)$ (maybe using inspection) Given 
$\cos(\pi/4 + 2\pi/3)$ 
and  
$\cos(\pi/4 + 4\pi/3)$
I wish to show that these values are given by $(\pm(\sqrt{6}) - \sqrt{2})/4$.
What is a quick way to arrive at these exact solutions (perhaps even by inspection)?
I am currently using the addition of cosine formula, but it is too slow. I'm thinking perhaps a graphic method can be used? 

Let $\,a=\cos \dfrac{\pi}{4} + i \sin \dfrac{\pi}{4}=\dfrac{1+i}{\sqrt{2}}\,$ and  $\,b=\cos \dfrac{2\pi}{3} + i \sin \dfrac{2\pi}{3}=\dfrac{-1+i\sqrt{3}}{2}\,$, so:
$$
ab = \dfrac{(1+i)(-1+i\sqrt{3})}{2 \sqrt{2}}=\dfrac{\left(-1-\sqrt{3}\right)+i\left(-1+\sqrt{3}\right)}{2 \sqrt{2}}
$$
Then $\,\cos\left(\dfrac{\pi}{4} + \dfrac{2 \pi}{3}\right) = \operatorname{Re}(ab) = \dfrac{-1-\sqrt{3}}{2\sqrt{2}}\,$.
