How to solve tan(x)= -1

First, note that $\tan({\pi\over4})=1$. Imagine this on the unit circle. Since we need to find $x$ when its tangent is negative 1, we have to get to the 2nd and 4th quadrant while retaining the tangent's absolute slope. To do that, we add $90^o$ or ${\pi\over2}$ and $270^o$ or ${3\pi\over2}$ to the angle ${\pi\over4}$. When we do that, we get the angles ${3\pi\over4}$ and ${7\pi\over4}$.

Q.E.D. 

tan(x) = -1        $x=\frac{3}{4}\pi + k*\pi$           $k\epsilon Z$