A) \[2\,\cos \,2\theta \]
B) \[\cos \,\,2\theta \]
C) \[2\,\sin \,2\theta \]
D) \[\sin \,\,2\theta \]
Correct Answer: A
Solution :
\[\frac{m}{n}=\frac{\tan \,({{120}^{o}}+\theta )}{\tan \,(\theta -{{30}^{o}})}\] \[\Rightarrow \,\,\frac{m+n}{m-n}=\frac{\tan \,(\theta +{{120}^{o}})+\tan \,(\theta -{{30}^{o}})}{\tan \,(\theta +{{120}^{o}})-\tan \,(\theta -{{30}^{o}})}\] (By componendo and dividendo) \[=\frac{\sin (\theta +{{120}^{o}})\cos (\theta -{{30}^{o}})+\cos (\theta +{{120}^{o}})\sin (\theta -{{30}^{o}})}{\sin (\theta +{{120}^{o}})\cos (\theta -{{30}^{o}})-\cos (\theta +{{120}^{o}})\sin (\theta -{{30}^{o}})}\] \[=\frac{\sin \,(2\theta +{{90}^{o}})}{\sin \,({{150}^{o}})}=\frac{\cos \,2\theta }{1/2}=2\,\cos \,2\theta \].
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