[tex]\displaystyle sin8x+sin10x+cosx=0\\\\2sin(\frac{8x+10x}{2})*cos(\frac{8x-10x}{2})+cosx=0\\\\2sin9x*cos(-x)+cosx=0 \\\\cosx(2sin9x+1)=0\\\\cosx=0; x=\frac{\pi }{2}+\pi n; n\in Z\\\\sin9x=-\frac{1}{2}\\\\9x=-\frac{\pi }{6}+2\pi n; x=-\frac{\pi }{54}+\frac{2\pi n}{9}; n\in Z\\\\9x=-\frac{5\pi }{6}+2\pi n; x=-\frac{5\pi}{54}+\frac{2\pi n}{9}; n\in Z[/tex]