\(\cos\left(2x\right)-\sin\left(3x\right)+\cos\left(5x\right)=\sin\left(10x\right)+\cos\left(8x\right)\)

\(\Leftrightarrow\cos\left(2x\right)-\cos\left(8x\right)+\cos\left(5x\right)-\sin\left(3x\right)-\sin\left(10x\right)=0\)

\(\Leftrightarrow-\left(\cos\left(8x\right)-\cos\left(2x\right)\right)+\cos\left(5x\right)-\left(\sin(10x\right)+\sin\left(3x\right))=0\)

\(\Leftrightarrow2\sin\left(5x\right)\sin\left(3x\right)+\cos\left(5x\right)-\sin\left(3x\right)-2\sin\left(5x\right)\cos\left(5x\right)=0\)

\(\Leftrightarrow2\sin\left(5x\right)(\sin\left(3x\right)-cos\left(5x\right))-\left(sin\left(3x\right)-cos\left(5x\right)\right)=0\)

\(\Leftrightarrow\left(2sin\left(5x\right)-1\right)\left(sin\left(3x\right)-cos\left(5x\right)\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}sin\left(5x\right)=\dfrac{1}{2}\\sin\left(3x\right)=cos\left(5x\right)\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}sin\left(5x\right)=\dfrac{1}{2}\\sin\left(3x\right)=sin\left(\dfrac{\pi}{2}-5x\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left[{}\begin{matrix}x=\dfrac{\pi}{30}+\dfrac{k2\pi}{3}\\x=\dfrac{\pi}{6}+\dfrac{k2\pi}{5}\end{matrix}\right.\\\left[{}\begin{matrix}3x=\dfrac{\pi}{2}-5x+k2\pi\\3x=\pi-\dfrac{\pi}{2}+5x+k2\pi\end{matrix}\right.\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{30}+\dfrac{k2\pi}{5}\\x=\dfrac{\pi}{6}+\dfrac{k2\pi}{5}\\x=\dfrac{\pi}{16}+\dfrac{k\pi}{4}\\x=-\dfrac{\pi}{4}+k\pi\end{matrix}\right.\)

\(\sin\left(5x\right)+\sin\left(3x\right)+2\cos\left(x\right)=1+\sin\left(4x\right)\)

\(\Leftrightarrow2\sin\left(4x\right)\cos\left(x\right)-\sin\left(4x\right)+2\cos\left(x\right)-1=0\)

\(\Leftrightarrow\sin\left(4x\right)(2\cos\left(x\right)-1)+(2\cos\left(x\right)-1)=0\)

\(\Leftrightarrow(2\cos\left(x\right)-1)(\sin\left(4x\right)+1)=0\)

\(\Rightarrow\left[{}\begin{matrix}\cos\left(x\right)=\dfrac{1}{2}\\\sin\left(4x\right)=-1\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{\pi}{3}+k2\pi\\4x=\dfrac{-\pi}{2}+k2\pi\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=\pm\dfrac{\pi}{3}+k2\pi\\x=\dfrac{-\pi}{8}+k\dfrac{\pi}{2}\end{matrix}\right.\)