Let S ={θ ∈[0,2 π]: 8^2 sin^2 θ+8^2 cos^2 θ=16}. Then n ( S )+Σ_θ ∈ S ( sec ((π)/4+2 θ) cosec ((π)/4+2 θ)) is equal to:

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