If $\theta =\frac{\pi }{21}$  then $\frac{\sin 23\theta -\sin 7\theta }{\sin 2\theta +\sin 14\theta }$

$\theta =\frac{\pi }{21}\Rightarrow 21\theta =\pi$

                $=\frac{\sin \left(21\theta +2\theta \right)-\sin \left(21\theta -14\theta \right)}{\sin 2\theta +\sin 14\theta }$

                $=\frac{\sin \left(\pi +2\theta \right)-\sin \left(\pi -14\theta \right)}{\sin 2\theta +\sin 14\theta }$

                $=\frac{-\sin 2\theta -\sin 14\theta }{\sin 2\theta +\sin 14\theta }$

                $=-\frac{\left(\sin 2\theta +\sin 14\theta \right)}{\sin 2\theta +\sin 14\theta }=-1$