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