The values of $\theta$ satisfying $\sin 7\theta =\sin 4\theta -\sin \theta$ and 

$0<\theta <\frac{\pi }{2}$ are

We have

$\begin{array}{l}\sin 7\theta +\sin \theta -\sin 4\theta =0\\ \Rightarrow 2\sin 4\theta \cos 3\theta -\sin 4\theta =0\\ \Rightarrow \sin 4\theta (2\cos 3\theta -1)=0\Rightarrow \sin 4\theta =0,\cos 3\theta =\frac{1}{2}\end{array}$

Now, $\sin 4\theta =0\Rightarrow 4\theta =\pi \Rightarrow \theta =\frac{\pi }{4}$

and $\cos 3\theta =\frac{1}{2}\Rightarrow 3\theta =\frac{\pi }{3}\Rightarrow \theta =\frac{\pi }{9}$