| If \[2\sin \theta =\sec \theta ,\] what is the value of \[{{\sin }^{4}}\theta +{{\cos }^{4}}\theta ?\] |
A) 1
B) \[\frac{1}{2}\]
C) \[\frac{1}{4}\]
D) \[\frac{1}{8}\]
Correct Answer: B
Solution :
| \[2\sin \theta =\sec \theta \] |
| \[\Rightarrow \] \[2\sin \theta \cos \theta =1\] |
| \[\Rightarrow \] \[\sin 2\theta =1\]\[\Rightarrow \]\[\sin 2\theta =sin90{}^\circ \] |
| \[\Rightarrow \] \[2\theta =90{}^\circ \]\[\Rightarrow \]\[\theta =45{}^\circ \] |
| \[\therefore \] \[{{\sin }^{4}}\theta +{{\cos }^{4}}\theta ={{\left( \frac{1}{\sqrt{2}} \right)}^{4}}+{{\left( \frac{1}{\sqrt{2}} \right)}^{4}}=\frac{1}{2}\] |
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