Answer
The solution set is $$\{0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{3\pi}{2},\frac{7\pi}{4}\}$$
Work Step by Step
$$\tan4x=0$$ over interval $[0,2\pi)$
1) Find corresponding interval for $4x$
Interval $[0,2\pi)$ can be written as
$$0\le x\lt2\pi$$
That means, for $4x$, the interval would be
$$0\le4x\lt8\pi$$
or $$4x\in[0,8\pi)$$
2) Now consider back the equation $$\tan4x=0$$
Over the interval $[0,8\pi)$, there are 8 values whose $\tan$ equals $0$, which are $0,\pi,2\pi,3\pi,4\pi,5\pi,6\pi,7\pi$, meaning that
$$4x=\{0,\pi,2\pi,3\pi,4\pi,5\pi,6\pi,7\pi\}$$
So $$x=\{0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{3\pi}{2},\frac{7\pi}{4}\}$$
