Let $\mathrm{f}\left(\mathrm{\theta }\right)=\frac{\cot \mathrm{\theta }}{1+\cot \mathrm{\theta }}\text{ and }\mathrm{\alpha }+\mathrm{\beta }=\frac{5\mathrm{\pi }}{4}$ then the value of $\mathrm{f}\left(\mathrm{\alpha }\right)\mathrm{f}\left(\mathrm{\beta }\right)$ is

$\begin{array}{c}\mathrm{f}\left(\mathrm{\beta }\right)=\mathrm{f}\left(\frac{5\mathrm{\pi }}{4}-\mathrm{\alpha }\right)=\frac{\cot \left(\frac{5\mathrm{\pi }}{4}-\mathrm{\alpha }\right)}{1+\cot \left(\frac{5\mathrm{\pi }}{4}-\mathrm{\alpha }\right)}\\ =\frac{1}{1+\tan \left(\frac{5\mathrm{\pi }}{4}-\mathrm{\alpha }\right)}\\ =\frac{1}{1+\frac{1-\tan \mathrm{\alpha }}{1+\tan \mathrm{\alpha }}}=\frac{1+\tan \mathrm{\alpha }}{2}\end{array}$

$\begin{array}{l}\mathrm{as}, \mathrm{f}\left(\mathrm{\alpha }\right)=\frac{\cot \mathrm{\alpha }}{1+\cot \mathrm{\alpha }}=\frac{1}{1+\tan \mathrm{\alpha }}\mathrm{we} \mathrm{have}\\ \mathrm{f}\left(\mathrm{\alpha }\right)\mathrm{f}\left(\mathrm{\beta }\right)=\frac{1}{2}\end{array}$