Answer
$\dfrac{\sqrt{2}}{2}$
Work Step by Step
$\cos{(\theta+ 2 \pi k}) = \cos{\theta}$
$\therefore \cos{\left(\dfrac{\pi}{4}+ (2 \pi \times 4) \right)} = \cos{\left(\dfrac{\pi}{4} \right)}$
$\cos{\left(\dfrac{\pi}{4} \right) }= \boxed{\dfrac{\sqrt{2}}{2}}$