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

If \(\tan A-\tan B=x\) and \(\cot A-\cot B=y\), then \(\cot (A-B)=\)
  1. \(\frac{x y}{x+y}\)
  2. \(\frac{x y}{x-y}\)
  3. \(\frac{x-y}{x y}\)
  4. \(\frac{y-x}{x y}\)

Solution

Since, \(\tan A-\tan B=x\) \(\Rightarrow \frac{1}{\cot A}-\frac{1}{\cot B}=x \Rightarrow \frac{\cot B-\cot A}{\cot A \cot B}=x\) \(\because \quad \cot A-\cot B=y\) (given) So, \(\quad \cot A \cot B=-\frac{y}{x}\) \(\because \cot (A-B)=\frac{\cot A \cot B+1}{\cot B-\cot A}=\frac{-\frac{y}{x}+1}{-y}=\frac{y-x}{x y}\) Hence, option (d) is correct.

Asked in: AP EAMCET 2019 (23 Apr Shift 1)

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