A Common Factor

${\tan\left(x\right)}^{2}-\tan\left(x\right) = 0$ holds tan x twice, so moving terms cannot isolate it. Read tan x as one letter u: ${u}^{2}-u = 0$, which factors as $u\left(u-1\right) = 0$.

Put tan x back: $\tan\left(x\right)\left(\tan\left(x\right)-1\right) = 0$. A product is zero only when a factor is, so tan x = 0 or tan x = 1. The first gives 0° and 180°, the second 45° and 225°: all four together.