If $A+B=225°$ then $\frac{\cot A}{1+\cot A}.\frac{\cot B}{1+\cot B}=$

$\Rightarrow$ $\cot \left(A+B\right)=\cot 225$

$\Rightarrow$ $\frac{\cot A.\cot B-1}{\cot B+\cot A}=\cot \left(180+45\right)$

$=\cot 45$

$=1$

$\Rightarrow$ $\cot A\cot B-1=\cot B+\cot A$

$\Rightarrow$ $\cot A\cot B=1+\cot A+\cot B\to \left(1\right)$

$\Rightarrow$ $\frac{\cot A.\cot B}{\left(1+\cot A\right)\left(1+\cot B\right)}=\frac{\cot A.\cot B}{\left(1+\cot A+\cot B\right)+\cot A.\cot B}$

$=\frac{\cot A.\cot B}{\cot A.\cot B+\cot A.\cot B}\left(\because \text{Eqn}\text{.}\left(1\right)\right)$

$\Rightarrow \frac{\cot A.\cot B}{2\cot A.\cot B}=\frac{1}{2}$