Question

If $\tan A-\tan B=x$ and $\cot B-\cot A=y$ then find the value of $\cot (A-B)$

Answer 1

$\tan A-\tan B=x$
$\Rightarrow \frac{1}{\cot A}-\frac{1}{\cot B}=x$
$\Rightarrow \frac{\cot B-\cot A}{\cot A-\cot B}=x$
$\Rightarrow \frac{y}{x}=\cot A\cot B$
Now
$\cot (A-B)=\frac{\cot A\cot B-1}{\cot A-\cos B}$
$=\frac{\frac{y}{x}-1}{-y}=\frac{y-x}{-xy}=\frac{x-y}{xy}$