Question

If $\tan \,A-\tan \,B=x$ and $\cot \,A-\cot \,B=y$ then $\cot \,(A-B)=$

• A. $\frac{x y}{x+y}$
• B. $\frac{x y}{x-y}$
• C. $\frac{x-y}{x y}$
• D. $\frac{y-x}{x y}$

Answer 1

Answer: (D)

Since, $\tan\, A-\tan \,B=x$
$\Rightarrow \, \frac{1}{\cot A}-\frac{1}{\cot B}=x$
$ \Rightarrow \,\frac{\cot \,B-\cot \,A}{\cot \,A \cot \,B}=x$
$\because \,\cot \,A-\cot \,B=y$ (given)
So, $ \cot \,A \cot \,B=-\frac{y}{x}$
$\because \,\cot (A-B)=\frac{\cot\, A \cot \,B+1}{\cot \,B-\cot \,A}$
$=\frac{-\frac{y}{x}+1}{-y}=\frac{y-x}{x y}$