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} $