Question

If $8cos2\theta +8sec2\theta =65,0<\theta <\frac{\pi }{2},$ then the value of $4cos4\theta$ is equal to

• A. $\frac{-23}{8}$
• B. $\frac{-31}{8}$
• C. $\frac{-31}{32}$
• D. $\frac{-33}{32}$
• E. $\frac{-32}{4}$

Answer 1

Answer: (B)

$8cos2\theta +8\sec 2\theta =65,\theta \in \left(0,\frac{\pi }{2}\right)$
$\Rightarrow$ $8\cos 2\theta +\frac{8}{\cos 2\theta }=65$
$\Rightarrow$ $8{\cos }^{2}2\theta -65\cos 2\theta +8=0$
$\Rightarrow$ $8{\cos }^{2}2\theta -64\cos 2\theta -\cos 2\theta +8=0$
$\Rightarrow$ $8\cos 2\theta (\cos 2\theta -8)-1(\cos 2\theta -8)=0$
$\Rightarrow$ $(\cos 2\theta -8)(8\cos 2\theta -1)=0$
$\Rightarrow$ $\cos 2\theta =\frac{1}{8},8$
$\Rightarrow$ ${\cos }^{2}2\theta =\frac{1}{64},64$
$\Rightarrow$ $2{\cos }^{2}2\theta =\frac{1}{32},128$
$\Rightarrow$ $(2{\cos }^{2}2\theta -1)=\left(\frac{1}{32}-1\right),(128-1)$
$\Rightarrow$ $\cos 4\theta =\frac{-31}{32},127$
$\Rightarrow$ $4\cos 4\theta =\frac{-31}{8},508$