A) $ \frac{3}{4} $
B) $ \frac{10}{3} $
C) $ 3\frac{3}{4} $
D) $ 8 $
Correct Answer: C
$ \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} $