solving:$\sqrt{\frac{1+\cos 4x}{2}}=\frac{1}{\sqrt{2}}$ How to find the values of $x$ such that $$\sqrt{\frac{1+\cos 4x}{2}}=\frac{1}{\sqrt{2}}$$

Sqaure both sides:
$$\frac{1+\cos(4x)}{2}=\frac{1}{2}$$
Multiply by two 
$$1+\cos(4x)=1$$
subtract one from both sides:
$$\cos(4x)=0$$
$\cos(y)=0$ when $y=\frac{\pi}{2}+ n\pi$ for $n\in \mathbb{Z}$. So we have to satisfy $4x=\frac{\pi}{2}+ n\pi$ for $n\in \mathbb{Z}$. So your solution for $x$ should be $x=\frac{\pi}{8}+\frac{n\pi}{4}$ where $n\in \mathbb{Z}$.
