Divide both sides by [tex]\dfrac12\cos x[/tex]:
[tex]\dfrac{\cos x}{\frac12\cos x}>\dfrac{\frac12\sin x}{\frac12\cos x}\implies2>\dfrac{\sin x}{\cos x}=\tan x[/tex]
If [tex]-\dfrac\pi2<x<\dfrac\pi2[/tex], then
[tex]\tan x<2\implies-\dfrac\pi2<x<\tan^{-1}2[/tex]
and more generally, for any integer [tex]n[/tex],
[tex]n\pi-\dfrac\pi2<x<n\pi+\tan^{-1}2[/tex]
