If \(y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^2 \alpha}\right)+\frac{1}{\sin ^2 \alpha}}\) where \(\alpha \in\left(\frac{3 \pi}{4}, \pi\right)\), then \(\frac{d y}{d \alpha}\ \rm at\ \alpha=\frac{5 \pi}{6}\) is
1
\(-\frac{1}{4}\)
2
\(\frac{4}{3}\)
3
4
4
-4