Question

If $\theta =\frac{\pi }{21}$, the value of $\frac{\sin 23\theta -\sin 7\theta }{\sin 2\theta +\sin 14\theta }=$

Answer 1

Given that,

$\frac{\sin 23\theta -\sin 7\theta }{\sin 2\theta +\sin 14\theta }$

$\begin{array}{c}=\frac{\sin 23\times \frac{\pi }{2}-\sin 7\times \frac{\pi }{2}}{\sin 2\times \frac{\pi }{2}+\sin 14\times \frac{\pi }{2}}\\ =\frac{\sin \frac{4140}{2}-\sin \frac{1260}{2}}{\sin \frac{360}{2}+\sin \frac{2520}{2}}\\ =\frac{\sin 2070-\sin 630}{\sin 180+\sin 1260}\end{array}$

$=\frac{\sin 1440}{\sin 1440}=1$