$_{\operatorname{Sin}(\frac{\pi }{2}+x)=\operatorname{Cos}x,\operatorname{Sin}(\pi +x)=-\operatorname{Sin}x\Rightarrow \frac{1}{2}-\operatorname{Sin}x\operatorname{Cos}x=0\xrightarrow{\times 2}1-\underbrace{2\operatorname{Sin}x\operatorname{Cos}x}_{\operatorname{Sin}2x}=0\Rightarrow \operatorname{Sin}2x=1\to 2x=2k\pi +\frac{\pi }{2}\Rightarrow x=k\pi +\frac{\pi }{4}}^{\operatorname{Sin}(\frac{5\pi }{6})=\operatorname{Sin}(\pi -\frac{\pi }{6})=\operatorname{Sin}\frac{\pi }{6}=\frac{1}{2}}$