Sin x + sin 5x +cos x+ cos 5x=0

[latex]sin x + sin 5x +cos x+ cos 5x=0\\(sinx+sin5x)+(cosx+cos5x)=0\\2sin3xcos2x+2cos3xcos2x=0\\2cos2x(sin3x+cos3x)=0\\\\cos2x=0\\2x=\frac{\pi}{2}+\pi n\\x=\frac{\pi}{4}+\frac{\pi n}{2}, \; n\in Z;\\\\sin3x+cos3x=0|:cos3x, \; cos3x \neq 0\\tg3x+1=0\\tg3x=-1\\3x=-\frac{\pi}{4}+\pi n\\x=-\frac{\pi}{12}+\frac{\pi n}{3}, \; n\in Z.[/latex]