Question

Let
$S =\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^2 \theta}+8^{2 \cos ^2 \theta}=16\right\} .$ Then
$n ( S )+\displaystyle\sum_{\theta \in S }\left(\sec \left(\frac{\pi}{4}+2 \theta\right) \operatorname{cosec}\left(\frac{\pi}{4}+2 \theta\right)\right)$
is equal to:

• A. 0
• B. $-2$
• C. $-4$
• D. 12

Answer 1

Answer: (C)

$8^{2 \sin ^2 \theta}+8^{2-2 \sin ^2 \theta}=16$
$y +\frac{64}{ y }=16$
$\Rightarrow y =8$
$\Rightarrow \sin ^2 \theta=1 / 2$
$n ( S )+\displaystyle\sum_{\theta \in S } \frac{1}{\cos (\pi / 4+2 \theta) \sin (\pi / 4+2 \theta)}$
$=4+(-2) \times 4=-4$