A) \[\frac{3}{4}\]
B) \[\frac{10}{3}\]
C) \[3\frac{3}{4}\]
D) \[3\]
Correct Answer: C
Solution :
| [c] \[\tan \theta \cdot \tan 2\theta =1\] \[\Rightarrow \] \[\tan \theta \frac{2\tan \theta }{1-{{\tan }^{2}}\theta }=1\] \[\Rightarrow \] \[2{{\tan }^{2}}=1-{{\tan }^{2}}\theta \] \[\Rightarrow \] \[3{{\tan }^{2}}\theta =1\] \[\Rightarrow \] \[\tan \theta =\frac{1}{\sqrt{3}}\] \[\Rightarrow \] \[\theta =30{}^\circ \] Now, \[{{\sin }^{2}}2\theta +{{\tan }^{2}}2\theta \] \[={{(sin60{}^\circ )}^{2}}+{{(tan60{}^\circ )}^{2}}\] \[={{\left( \frac{\sqrt{3}}{2} \right)}^{2}}+{{(\sqrt{3})}^{2}}=\frac{3}{4}+3\] \[=\frac{3+12}{4}=\frac{15}{4}=3\frac{3}{4}\] |
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