A) \[\frac{-23}{8}\]
B) \[\frac{-31}{8}\]
C) \[\frac{-31}{32}\]
D) \[\frac{-33}{32}\]
E) \[\frac{-32}{4}\]
Correct Answer: B
Solution :
\[8cos2\theta +8\sec 2\theta =65,\theta \in \left( 0,\frac{\pi }{2} \right)\] \[\Rightarrow \] \[8\cos 2\theta +\frac{8}{\cos 2\theta }=65\] \[\Rightarrow \] \[8{{\cos }^{2}}2\theta -65\cos 2\theta +8=0\] \[\Rightarrow \] \[8{{\cos }^{2}}2\theta -64\cos 2\theta -\cos 2\theta +8=0\] \[\Rightarrow \] \[8\cos 2\theta (\cos 2\theta -8)-1(\cos 2\theta -8)=0\] \[\Rightarrow \] \[(\cos 2\theta -8)(8\cos 2\theta -1)=0\] \[\Rightarrow \] \[\cos 2\theta =\frac{1}{8},8\] \[\Rightarrow \] \[{{\cos }^{2}}2\theta =\frac{1}{64},64\] \[\Rightarrow \] \[2{{\cos }^{2}}2\theta =\frac{1}{32},128\] \[\Rightarrow \] \[(2{{\cos }^{2}}2\theta -1)=\left( \frac{1}{32}-1 \right),(128-1)\] \[\Rightarrow \] \[\cos 4\theta =\frac{-31}{32},127\] \[\Rightarrow \] \[4\cos 4\theta =\frac{-31}{8},508\]
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