$\text{ The range of the function, }f\left(x\right)={\cot }^{-1}{\log }_{0.5}\left(x^4-2x^2+3\right) is$

$$\begin{gathered} x^4-2 x^2+3 = {\left(x^2-1\right)}^2+2 \ge 2 \\ \Rightarrow \log {}_{0.5}\left(x^4-2x^2+3\right) \le \log {}_{0.5}2 =-\log {}_{2}2=-1 \left( \sin ce x\ge y \Rightarrow \log {}_{a}x \le \log {}_{a}y if 0<a<1\right) \end{gathered}$$

$\Rightarrow co{t}^{-1 }\left(\log {}_{0.5}\left(x^4-2x^2+3\right)\right) \ge co{t}^{-1}\left(-1\right) \left( \sin ce co{t}^{-1}x fumction is decrea\sin g \right)$

$$\begin{gathered} \Rightarrow y= co{t}^{-1}\left(\log {}_{0.5}\left(x^4-2x^2+3\right)\right) \ge \mathrm{\pi } -{\cot }^{-1}1=\frac{3\mathrm{\pi }}{4} \\ \Rightarrow \mathrm{y} \in [ \frac{3\mathrm{\pi }}{4},\mathrm{\pi }) \left(\mathrm{since} {\cot }^{-1}\mathrm{x} \in \left(0,\mathrm{\pi }\right)\right) \end{gathered}$$