Find the number of integers for which tan|x|=|tanx|.$x\in(-2\pi,2\pi)$
I used desmos.com and found the following x coordinate where the curve y=tan|x|and
y=|tanx|intersect. The points are at x=0,+1,-1,+4,-4. Hence the number of integers where it intersect is 5. Please help me with finding the correct answer I am not sure about it.
Outline/Hint: First, there's an obvious solution at $x=0$. Then since $\tan|x|$ is even you just need to find solutions in the interval $(0,2\pi)$. Notice that on this interval $\tan\!|x| = |\tan(x)|$ exactly when $\tan(x)$ is positive. On which intervals is $\tan(x)$ positive? Then which of the numbers $\{1,2,3,4,5,6\}$ live in these intervals?