Verified answer

[tex]\displaystyle\bf\\Cosx+Sin\frac{x}{2} =0\\\\\\1-2Sin^{2} \frac{x}{2} +Sin\frac{x}{2} =0\\\\\\2Sin^{2} \frac{x}{2} -Sin\frac{x}{2} -1=0\\\\\\Sin\frac{x}{2} =m \ \, ; \ -1\leq m\leq 1\\\\\\2m^{2} -m-1=0\\\\D=(-1)^{2} -4\cdot 2\cdot(-1)=1+8=9=3^{2} \\\\\\m_{1} =\frac{1-3}{4} =-\frac{1}{2} \\\\\\m_{2} =\frac{1+3}{4} =1[/tex]

[tex]\displaystyle\bf\\1)\\\\Sin\frac{x}{2} =-\frac{1}{2} \\\\\\\frac{x}{2} =\Big(-1\Big)^{n} arcSin\Big(-\frac{1}{2} \Big)+\pi n,n\in Z\\\\\\\frac{x}{2} =\Big(-1\Big)^{n+1} arcSin\frac{1}{2} +\pi n,n\in Z\\\\\\\frac{x}{2} =\Big(-1\Big)^{n+1}\frac{\pi }{6} +\pi n,n\in Z\\\\\\x =\Big(-1\Big)^{n+1}\frac{\pi }{3} +2\pi n,n\in Z\\\\\\2)\\\\Sin\frac{x}{2} =1\\\\\\\frac{x}{2} =\frac{\pi }{2} +2\pi n,n\in Z\\\\\\x=\pi +4\pi n,n\in Z[/tex]

[tex]\displaystyle\bf\\Otvet \ : \ (-1)^{n+1} \frac{\pi }{3} +2\pi n \ \ ; \ \ \pi +4\pi n \ , \ n\in Z[/tex]