Question

$(\sin 3x+\sin x)\sin x+(\cos 3x-\cos x)\cos x=0$

Answer 1

$\begin{array}{c}\left(\sin 3x+\sin x\right)\sin x+\left(\cos 3x-\cos x\right)\cos x\\ \left(3\sin x-4\sin x+\sin x\right)\sin x+\left(4{\cos }^{3}x-3\cos x-\cos x\right)\cos x\\ \left(4\sin x-4{\sin }^{3}x\right)\sin x+\left(4{\cos }^{3}x-4\cos x\right)\cos x\\ 4{\sin }^{2}x-4{\sin }^{4}x+4{\cos }^{4}x-4{\cos }^{2}x\\ 4{\sin }^{2}x-4{\cos }^{2}x-4\left({\sin }^{4}x-{\cos }^{4}x\right)\\ 4\left({\sin }^{2}x-4{\sin }^{2}x\right)-4\left({\sin }^{2}x+{\cos }^{2}x\right)\left({\sin }^{2}x-{\cos }^{2}x\right)\\ 4\left({\sin }^{2}x-4{\cos }^{2}x\right)-4\left({\sin }^{2}x-{\cos }^{2}x\right)\\ =0\end{array}$