Answer
$ \dfrac{-\sqrt{6}-\sqrt{2}}{4}$
Work Step by Step
Addition formula
$$\cos{(A+B)}= \cos{A} \cos{B}-\sin{A}\sin{B}$$
$\therefore \cos{\left(\dfrac{\pi}{4}+\dfrac{2\pi}{3} \right)} = \cos{\left(\dfrac{\pi}{4} \right)} \cos{\left(\dfrac{2\pi}{3} \right)} - \sin{\left(\dfrac{\pi}{4}\right)} \sin{\left(\dfrac{2\pi}{3} \right)}$
$\cos{\left(\dfrac{\pi}{4}+\dfrac{2\pi}{3} \right)} = \dfrac{\sqrt{2}}{2} \times \dfrac{-1}{2} -\dfrac{\sqrt{2}}{2} \times \dfrac{\sqrt{3}}{2}$
$\cos{\left(\dfrac{\pi}{4}+\dfrac{2\pi}{3} \right)} = \dfrac{-\sqrt{6}-\sqrt{2}}{4}$
