Let S = \ θ ∈ [0, 2π]: 8 ^ 2 ^ 2 θ + 8 ^ 2 ^ 2 θ = 16\ . Then n(S)...

Given, ${8}^{2\sin^{2}\theta} + {8}^{2\cos^{2}\theta} = 16$

$⇒ {8}^{2\sin^{2}\theta} + {8}^{2 - 2\sin^{2}\theta} = 16$

Now let ${8}^{2\sin^{2}\theta} = y$

$⇒ y + \dfrac{64}{y} = 16$

$⇒ y = 8$

$⇒ {8}^{2\sin^{2}\theta} = 8$

$⇒ \sin^{2}\theta = \dfrac{1}{2}$

$⇒ \theta \in \left\{\dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}\right\}$

Now $n\left(S\right) + \mathop{∑}_{\theta \in S}\left(\sec \left(\dfrac{\pi}{4} + 2\theta\right) cosec \left(\dfrac{\pi}{4} + 2\theta\right)\right)$

$= n\left(S\right) + \mathop{∑}_{\theta \in S}\dfrac{1}{\cos \left(\dfrac{\pi}{4} + 2\theta\right)\sin \left(\dfrac{\pi}{4} + 2\theta\right)}$

$= 4 + \mathop{∑}_{\theta \in S}\dfrac{2}{2\cos \left(\dfrac{\pi}{4} + 2\theta\right)\sin \left(\dfrac{\pi}{4} + 2\theta\right)}$

$= 4 + \mathop{∑}_{\theta \in S}2cos\text{ec}\left(\dfrac{\pi}{2} + 4\theta\right)$

$= 4 + 2cos\text{ec}\left(\dfrac{\pi}{2} + \pi\right)\text{+}2cos\text{ec}\left(\dfrac{\pi}{2} + 3\pi\right)\text{+}2cos\text{ec}\left(\dfrac{\pi}{2} + 5\pi\right)\text{+}2cos\text{ec}\left(\dfrac{\pi}{2} + 7\pi\right)$

$= 4 + 2\left[- cos\text{ec}\dfrac{\pi}{2} - cos\text{ec}\dfrac{\pi}{2} - cos\text{ec}\dfrac{\pi}{2} - cos\text{ec}\dfrac{\pi}{2}\right]$

$= 4 + \left(- 2\right) \times 4 = - 4$