Question

2√3cos2·(3π/2+x/2)=sinx​

Answer 1

$2\sqrt{3} Cos^{2}(\frac{3\pi }{2}+\frac{x}{2})=Sinx\\\\2\sqrt{3}Sin^{2} \frac{x}{2} -Sinx=0\\\\2\sqrt{3}Sin^{2} \frac{x}{2} -2Sin\frac{x}{2}Cos\frac{x}{2}=0\\\\\sqrt{3}Sin^{2} \frac{x}{2} -Sin\frac{x}{2}Cos\frac{x}{2}=0\\\\Sin\frac{x}{2}(\sqrt{3}Sin\frac{x}{2}-Cos\frac{x}{2})=0\\\\1)Sin\frac{x}{2}=0\\\\\frac{x}{2}=\pi n,n\in Z\\\\\boxed{x=2\pi n,n\in Z}\\\\2)\sqrt{3}Sin\frac{x}{2}-Cos\frac{x}{2}=0|:Cos\frac{x}{2}\neq 0\\\\\sqrt{3}tg\frac{x}{2} -1=0$

$tg\frac{x}{2}=\frac{1}{\sqrt{3}}\\\\\frac{x}{2}=arctg\frac{1}{\sqrt{3} }+\pi n,n\in Z\\\\\frac{x}{2}=\frac{\pi }{6}+\pi n,n\in Z\\\\\boxed{x=\frac{\pi }{3}+2\pi n,n\in Z}$