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  4. If 8 text cos text 2θ +8 text sec text 2θ =65, text 0<θ <(π /2), then the value of 4 text cos text 4θ is equal to

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Q. If $ 8\text{ }cos\text{ }2\theta +8\text{ }sec\text{ }2\theta =65,\text{ }0<\theta <\frac{\pi }{2}, $ then the value of $ 4\text{ }cos\text{ }4\theta $ is equal to

KEAMKEAM 2011

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 $