$_{(\operatorname{Sin}x-\tan x)\tan (\frac{3\pi }{2}-x)=\operatorname{Cos}\frac{4\pi }{3}\Rightarrow (\operatorname{Sin}x-\tan x)Cotx=-\frac{1}{2}\Rightarrow \operatorname{Sin}xCotx-\underbrace{\tan xCotx}_{1}=-\frac{1}{2}\Rightarrow \operatorname{Sin}x\times \frac{\operatorname{Cos}x}{\operatorname{Sin}x}=\frac{1}{2}\Rightarrow \operatorname{Cos}x=\frac{1}{2}=\operatorname{Cos}\frac{\pi }{3}\Rightarrow x=2k\pi \pm \frac{\pi }{3}}^{\tan (\frac{3\pi }{2}-x)=Cotx,\operatorname{Cos}\frac{4\pi }{3}=\operatorname{Cos}(\pi +\frac{\pi }{3})=-\operatorname{Cos}\frac{\pi }{3}=-\frac{1}{2}}$