\(\displaystyle{\sin{{x}}}+{\sin{{3}}}{x}={2}\frac{{\sin{{\left({3}{x}+{x}\right)}}}}{{2}}\times\frac{{\cos{{\left({3}{x}-{x}\right)}}}}{{2}}={0}\)

\(\displaystyle{2}{\sin{{2}}}{x}{\cos{{x}}}={0}\)

\(\displaystyle{\sin{{2}}}{x}={0}\) or \(\cos x=0\)

if \(\displaystyle{\sin{{2}}}{x}={0}\), then

\(\displaystyle{2}{x}={0}{\quad\text{or}\quad}\pi{\quad\text{or}\quad}{2}\pi\)

\(\displaystyle{x}={0},\frac{\pi}{{2}},\pi\)

if \(\displaystyle{\cos{{x}}}={0}\) then

\(\displaystyle{x}=\frac{\pi}{{2}},{3}\frac{\pi}{{2}}\)