The solution set of $\sin 7x+\sin 3x=\sin 2x+\sin 8x$  is $\left(n\in z\right)$

$2\sin 5x.\cos 2x=2\sin 5x.\cos 3x$

$2\sin 5x.\cos 2x-2\sin 5x.\cos 3x=0$

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

$-4\sin 5x\sin \frac{5x}{2}\sin \frac{x}{2}=0$

$\sin 5x=0$  (or) $\sin \frac{5x}{2}=0$  (or) $\sin \frac{x}{2}=0$

$5x=n\pi$  (or)   $\frac{5x}{2}=n\pi$  (or)    $\frac{x}{2}=n\pi$

$x=\frac{n\pi }{5}$    (or)    $x=\frac{2n\pi }{5}$    (or)   $x=2n\pi$

Common solution $x=\frac{n\pi }{5}$