Finding all $x$ such that $|\tan x | \leq 2\sin x$

Finding all $x$ such that $|\tan x | \leq 2\sin x$

I need to find all real numbers x that satisfy: $$|\tan x | \leq 2\sin x
\;\; and\;\; x \in [ -\pi, \pi]$$ in terms of unions of intervals.
I know it's equivalent to: $-2\sin x \leq \tan x \leq 2 \sin x $
I tried dividing into cases where $\sin x = 0 $ or $\sin x \neq0$ .
and also, $\cos x \gt 0 $ or $\cos x \lt 0 $. But alas, my attempts to
solve this failed.
Perhaps I'm missing something?
In addition, I'm struggling with these types of questions (finding
solution sets) as it's hard for me to see whether the attempted solution
shows a double inclusion or a single one. In other words whether each step
of inference is an equivalence or just an implication.
Thanks.