Let \( S=\left\{ \theta \in [0,2\pi] : 8^{2\sin^2\theta}+8^{2\cos^2\theta}=16 \right\}. \) Then \( n(S)+\displaystyle\sum_{\theta \in S}\big(\sec \left(\frac{\pi}{4}+2\theta\right)\csc \left(\frac{\pi}{4}+2\theta\right)\big) \) is equal to:
\( 0 \)
\( -2 \)
\( -4 \)
\( 12 \)
