$\underline{k\in \mathbb{C}}$  

 

$a)$  

$\sin 3x-\sin 2x=\sin x$  

$\sin 3x-\sin x=\sin 2x$  

$\sin 3x-\sin x=2\sin \left(\frac{3x-x}{2}\right)\cos \left(\frac{3x+x}{2}\right)=2\sin x\cos 2x$  

$2\sin x\cos 2x=\sin 2x=2\sin x\cos x$  

$2\sin x\cos 2x-2\sin x\cos x=0$  

$2\sin x\left(\cos 2x-\cos x\right)=0$  

$\sin x=0$  

$\underline{x=k\pi }$   

 

$\cos 2x-\cos x=0$  

$\cos 2x-\cos x=2{\cos }^{2}x-1-\cos x=2{\cos }^{2}x-\cos x-1=0$  

 

$t=\cos x$   

$2t^2-t-1=0$  

$\Delta =1+8=9$  

$\sqrt{\Delta }=3$  

$t_1=\frac{1-3}{4}=-\frac{1}{2}$  

$t_2=\frac{1+3}{4}=1$  

 

$\cos x=-\frac{1}{2}=-\cos \left(x+\pi \right)$  

$\cos \left(x+\pi \right)=\frac{1}{2}$  

$x+\pi =\frac{\pi }{3}+2k\pi \vee x+\pi =2\pi -\frac{\pi }{3}+2k\pi$  

$\underline{x=-\frac{2}{3}\pi +2k\pi \vee x=\frac{2}{3}\pi +2k\pi }$         

 

$\cos x=1$  

$\underline{x=2k\pi }$  

  

$\text{Podsumowując:}x=k\pi \vee x=-\frac{2}{3}\pi +2k\pi \vee x=\frac{2}{3}\pi +2k\pi$  

 

$b)$  

$\cos 5x-\sin 3x=\cos x$  

$\cos 5x-\cos x=\sin 3x$  

$\cos 5x-\cos x=-2\sin \left(\frac{6x}{2}\right)\sin \left(\frac{4x}{2}\right)=-2\sin 3x\sin 2x=\sin 3x$  

$-2\sin 3x\sin 2x-\sin 3x=0$  

$-\sin 3x\left(2\sin 2x+1\right)=0$  

$\sin 3x=0$  

$3x=k\pi$  

$\underline{x=k\frac{\pi }{3}}$   

 

$2\sin 2x+1=0$  

$\sin 2x=-\frac{1}{2}=-\sin \left(\pi +2x\right)$   

$\sin \left(\pi +2x\right)=\frac{1}{2}$  

$\pi +2x=\frac{\pi }{6}+2k\pi \vee \pi +2x=\pi -\frac{\pi }{6}+2k\pi$   

$\underline{x=-\frac{5}{12}\pi +k\pi \vee x=-\frac{\pi }{12}+k\pi }$    

 

$c)$  

$\cos 6x+\sin 5x+\cos 2x=\sin 3x$  

$\cos 6x+\cos 2x+\sin 5x-\sin 3x=2\cos \left(\frac{8x}{2}\right)\cos \left(\frac{4x}{2}\right)+2\sin \left(\frac{2x}{2}\right)\cos \left(\frac{8x}{2}\right)=$  

$=2\cos 4x\cos 2x+2\sin x\cos 4x=0$  

$\cos 4x\left(2\cos 2x+2\sin x\right)=0$  

$\cos 4x=0\vee 2\cos 2x+2\sin x=0$    

$4x=\frac{\pi }{2}+k\pi$

$\underline{x=\frac{\pi }{8}+k\frac{\pi }{4}}$   

 

$2\cos 2x+2\sin x=0\mid :2$  

$\cos 2x+\sin x=1-2{\sin }^{2}x+\sin x=0$   

$\sin x=t$  

$-2t^2+t+1=0$  

$\Delta =1+8=9$  

$\sqrt{\Delta }=3$  

$t_1=\frac{-1-3}{-4}=1$  

$t_2=\frac{-1+3}{-4}=-\frac{1}{2}$  

 

$\sin x=1$  

$\underline{x=\frac{\pi }{2}+2k\pi }$   

 

$\sin x=-\frac{1}{2}=-\sin \left(\pi +x\right)$  

$\sin \left(\pi +x\right)=\frac{1}{2}$  

$\pi +x=\frac{\pi }{6}+2k\pi \vee \pi +x=\pi -\frac{\pi }{6}+2k\pi$  

$\underline{x=-\frac{5}{6}\pi +2k\pi \vee x=-\frac{\pi }{6}+2k\pi }$   

 

$d)$  

$\cos 7x-\sin 7x=\cos x-\sin x$  

$\cos 7x-\cos x+\sin x-\sin 7x=0$  

$\cos 7x-\cos x+\sin x-\sin 7x=-2\sin \left(\frac{8x}{2}\right)\sin \left(\frac{6x}{2}\right)-2\sin \left(\frac{6x}{2}\right)\cos \left(\frac{8x}{2}\right)=$   

$=-2\sin 4x\sin 3x-2\sin 3x\cos 4x=\sin 3x\left(-2\sin 4x-2\cos 4x\right)=0$   

$\sin 3x=0\vee -2\sin 4x-2\cos 4x=0$  

$3x=k\pi$  

$x=k\frac{\pi }{3}$  

 

$-2\sin 4x-2\cos 4x=0\iff \cos 4x+\sin 4x=0$   

$t=4x$  

$\sin t+\cos t=0$  

${\left(\sin t+\cos t\right)}^2=0$  

${\sin }^{2}t+{\cos }^{2}t+2\sin t\cos t=0$  

$1+2\sin 2t=0$  

$\sin 2t=-1$  

$2t=\frac{3}{2}\pi +2k\pi$  

$8x=\frac{3}{2}\pi +2k\pi$   

$x=\frac{3}{16}+k\frac{\pi }{4}$