This is not an identity but an equation. You're supposed to find the values of [tex]x[/tex] for which this is true. It is *not* true for all values of [tex]x[/tex].
To see why:
[tex]\tan x[/tex] has period [tex]\pi[/tex], which means [tex]\tan(x+\pi)=\tan x[/tex].
The angle sum identity for [tex]\sin x[/tex] says that
[tex]\sin(x+\pi)=\sin x\cos \pi+\cos x\sin\pi=-\sin x[/tex]
So
[tex]\tan(x+\pi)+2\sin(x+\pi)=\tan x-2\sin x[/tex]
By definition of [tex]\tan x[/tex],
[tex]\tan x=\dfrac{\sin x}{\cos x}[/tex]
and so
[tex]\tan x-2\sin x=\sin x\left(\dfrac1{\cos x}-2\right)[/tex]
which is only 0 if either [tex]\sin x=0[/tex] (which only happens for certain values of [tex]x[/tex]) or [tex]\dfrac1{\cos x}-2=0\implies\cos x=\dfrac12[/tex] (which also only happens for certain values of [tex]x[/tex]). It's these values of [tex]x[/tex] you want to find.
[tex]\sin x=0[/tex] whenever [tex]x[/tex] is a multiple of [tex]\pi[/tex], i.e. [tex]x=n\pi[/tex] for any integer [tex]n[/tex].
If [tex]0\le x<2\pi[/tex], then [tex]\cos x=\dfrac12[/tex] is true for [tex]x=\dfrac\pi3[/tex] and [tex]x=\dfrac{5\pi}3[/tex]. Then to account for all other possible values, we add a multiple of [tex]2\pi[/tex], so that [tex]x=\dfrac\pi3+2n\pi[/tex] or [tex]x=\dfrac{5\pi}3+2n\pi[/tex] for integers [tex]n[/tex].
