| If \[\sin \theta \cos \theta =\frac{\sqrt{3}}{2},\] then the value of \[{{\sin }^{4}}\theta +{{\cos }^{4}}\theta \] is |
A) \[\frac{7}{8}\]
B) \[\frac{5}{8}\]
C) \[\frac{3}{8}\]
D) \[\frac{1}{8}\]
Correct Answer: B
Solution :
| Given that, \[\sin \theta \cdot \cos \theta =\frac{\sqrt{3}}{4}\] (i) |
| Now, we have \[{{\sin }^{4}}\theta +{{\cos }^{4}}\theta \] |
| \[={{({{\sin }^{2}}\theta +{{\cos }^{2}}\theta )}^{2}}-2{{\sin }^{2}}\theta {{\cos }^{2}}\theta \] |
| \[={{(1)}^{2}}-2\,\,{{(\sin \theta \cos \theta )}^{2}}\] |
| \[=1-2{{\left( \frac{\sqrt{3}}{4} \right)}^{2}}=1-2\cdot \frac{3}{16}=1-\frac{3}{8}=\frac{5}{8}\] |
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