Given,
\(m\sin \theta = n \sin (\theta + 2\alpha)\)
\(\frac mn = \frac {\sin(\theta + 2 \alpha )}{\sin \theta}\)
Apply componendo and dividendo
\(\frac{m + n}{m-n} = \frac {\sin (\theta + 2\alpha) + \sin \theta}{\sin (\theta + 2\alpha) - \sin \theta}\)
\(\frac{m + n}{m-n} = \frac {\sin (\theta + \alpha) \cos \alpha}{\cos(\theta + \alpha) \sin \alpha}\)
\(\frac{m + n}{m-n} = \tan(\theta + \alpha) \cot \alpha\)
