Q75 / 180 · COMEDK 2018 · Mathematics

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

  • A
    $\frac{1}{y}-\frac{1}{x}$
  • B
    $\frac{1}{x}-\frac{1}{y}$
  • Correct answer
    C
    $\frac{1}{x}+\frac{1}{y}$
  • D
    None of these

Solution

We have, $$ \begin{aligned} \tan A-\tan B &=x \\ \text { and } \cot B-\cot A &=y \\ \Rightarrow \frac{1}{\tan B}-\frac{1}{\tan A} &=y \\ \Rightarrow \quad \frac{\tan A-\tan B}{\tan A \tan B} &=y \\ \Rightarrow \quad \tan A \tan B &=\frac{x}{y} \\ \Rightarrow \quad \cot A \cot B &=\frac{y}{x} \\ \text { Now, } \cot (A-B) &=\frac{1+\cot A \cot B}{\cot B-\cot A} \\ &=\frac{1+\frac{y}{x}}{y}=\frac{x+y}{x y}=\frac{1}{y}+\frac{1}{x} \end{aligned} $$

Hence, the correct answer is option: 'C'.

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