Let $\frac{3\pi }{4}<\theta <\pi$  and $\sqrt{2\cot \theta +\frac{1}{{\sin }^2\theta }}=k-\cot \theta$  then k is equal to

$G.E=\sqrt{2\cot \theta +\left(1+{\cot }^2\theta \right)}$

$=-\left(1+\cot \theta \right)$ , (since $\frac{3\pi }{4}<\theta <\pi$ ,  $\cot \theta <-1,\Rightarrow \left(1+\cot \theta <0\right)$ )

$=-1-\cot \theta$ , $k=-1$