Question

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

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

Answer 1

Answer: (B)

$8cos2\theta +8sec2\theta =65$, $0<\theta <\pi /2$
$\Rightarrow 8co{s}^{2}2\theta +8=65cos2\theta$
$\Rightarrow 8co{s}^{2}2\theta -65cos2\theta +8=0$
$\Rightarrow \left(cos2\theta -8\right)\left(8cos2\theta -1\right)=0$
$\Rightarrow cos2\theta =1/8$,
$cos2\theta =8$,
$\left(cos2\theta \ne 8\text{ as}cos2\theta \in \left[-1,1\right]\right)$
So,$cos2\theta =1/8$. Now, $4cos4\theta =4(2co{s}^{2}2\theta -1)$
$=4(2\cdot (1/64)-1)=-31/8$