Question

If $A+B={225}^{\circ },$ then find the value of $\frac{\cot A}{1+\cot A}\times \frac{\cot B}{1+\cot B}$

Answer 1

$A+B=225\Rightarrow \cot (A+B)=\cot 225$
$\Rightarrow \frac{\cot A\cot B-1}{\cot A+\cot B}=1$
$\Rightarrow \cot A\cot B-1=\cot A+\cot B$
Now
$\frac{\cot A}{1+\cot A}\times \frac{\cot B}{1+\cot B}=\frac{\cot A\cot B}{1+\cot A+\cot B+\cot A\cot B}$
$=\frac{\cot A\cot B}{1+\cot A+\cot B-1+\cot A\cot B}=\frac{1}{2}$