Let  $\mathrm{f}\left(\mathrm{\theta }\right)=\frac{1}{1+(\cot \mathrm{\theta }{)}^{\mathrm{x}}},\text{ and }\mathrm{S}=\sum _{\mathrm{\theta }=1^{\circ }}^{{89}^{\circ }} \mathrm{f}\left(\mathrm{\theta }\right)$  then the value of $\sqrt{5}$ is

we have $\mathrm{f}\left(\mathrm{\theta }\right)=\frac{(\sin \mathrm{\theta }{)}^{\mathrm{x}}}{(\cos \mathrm{\theta }{)}^{\mathrm{x}}+(\sin \mathrm{\theta }{)}^{\mathrm{x}}}$

$\begin{array}{l}\Rightarrow \mathrm{f}\left(\mathrm{\theta }\right)+\mathrm{f}\left(\frac{\mathrm{\pi }}{2}-\mathrm{\theta }\right)=1.\\ \therefore \mathrm{S}=\sum _{\mathrm{\theta }=1^{\circ }}^{890} \mathrm{f}\left(\mathrm{\theta }\right)=\mathrm{f}\left(1^{\circ }\right)+\mathrm{f}\left(2^{\circ }\right)+\dots +\mathrm{f}\left({88}^{\circ }\right)+\mathrm{f}\left({89}^{\circ }\right)\\ =\left(\underset{44\text{ times }}{\underbrace{1+1+1+\dots \dots +1}}\right)+\frac{1}{2}=44+\frac{1}{2}=\frac{89}{2}\end{array}$