If I have $\displaystyle tan{x}+1=0$, then I have $\displaystyle tan{x} = -1$

well, tan is equal to 1 in $\displaystyle \frac{\pi}{4}$. It's equal to -1 in quadrant 2 and 4.

So $\displaystyle x = \frac{3}{4}{\pi}+2n{\pi}$ where n is all reals.

but why isn't it also $\displaystyle x = \frac{7}{4}{\pi}+2n{\pi}$, since tan is negative in the 4th quadrant as well?