If

Solution:

(d) −1 −cot α

We have:

$\sqrt{2 \cot \alpha+\frac{1}{\sin ^{2} \alpha}}$

$=\sqrt{\frac{2 \cos \alpha}{\sin \alpha}+\frac{1}{\sin ^{2} \alpha}}$

 

$=\sqrt{\frac{2 \sin \alpha \cos \alpha+1}{\sin ^{2} \alpha}}$

$=\sqrt{\frac{2 \sin \alpha \cos \alpha+\sin ^{2} \alpha+\cos ^{2} \alpha}{\sin ^{2} \alpha}}$

 

$=\sqrt{\frac{(\sin \alpha+\cos \alpha)^{2}}{\sin ^{2} \alpha}}$

$=\sqrt{(1+\cot \alpha)^{2}}$

 

$=|1+\cot \alpha|$

$=-(1+\cot \alpha) \quad\left[\right.$ When $\left.\frac{3 \pi}{4}<\alpha<\pi, \cot \alpha<-1 \Rightarrow \cot \alpha+1<0\right]$

 

$=-1-\cot \alpha$