Quantitative Aptitude Ques 820

Question: If $ \tan \theta \cdot \tan 2\theta =1, $ then the value of $ {{\sin }^{2}}2\theta +{{\tan }^{2}}2\theta $ is equal to

Options:

A) $ \frac{3}{4} $

B) $ \frac{10}{3} $

C) $ 3\frac{3}{4} $

D) $ 8 $

Show Answer

Answer:

Correct Answer: C

Solution:

  • $ \tan \theta \cdot \tan 2\theta =1 $

$ \Rightarrow $ $ \tan 2\theta =\frac{1}{\tan \theta }=\cot \theta =\tan (90-\theta ) $

$ \Rightarrow $ $ 2\theta =90-\theta $
$ \Rightarrow $ $ 3\theta =90{}^\circ $ $ \theta =30{}^\circ $

$ \therefore $ $ {{\sin }^{2}}2\theta +{{\tan }^{2}}2\theta ={{\sin }^{2}}(60{}^\circ )+{{\tan }^{2}}(60{}^\circ ) $ $ ={{( \frac{\sqrt{3}}{2} )}^{2}}+{{(\sqrt{3})}^{2}} $ $ =\frac{3}{4}+3=\frac{15}{4}=3\frac{3}{4} $