Question

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

• A. $\frac{1}{y}-\frac{1}{x}$
• B. $\frac{1}{x}-\frac{1}{y}$
• C. $\frac{1}{x}+\frac{1}{y}$
• D. none of these

Answer 1

Answer: (C)

We have, $\tan A-\tan B=x$ and $\cot B-\cot A=y$
$\Rightarrow \frac{1}{\tan B}-\frac{1}{\tan A}=y$
$\frac{\tan A-\tan B}{\tan A\tan B}=y\Rightarrow \frac{x}{y}=\tan A\tan B$
$\cot \left(A-B\right)=\frac{1}{\tan \left(A-B\right)}=\frac{1}{\frac{\tan A-\tan B}{1+\tan A\tan B}}$
$=\frac{1}{\frac{x}{1+\frac{x}{y}}}=\frac{y+x}{xy}=\frac{1}{x}+\frac{1}{y}$