If tan A - tan B = x and cot A - cot B = y, then cot (A - B) = ?1. x -

1.	If tan A - tan B = x and cot A - cot B = y, then cot (A - B) = ?1. x - y2. \(\rm \frac1x+\frac1y\)3. \(\rm \frac1x-\frac1y\)4. \(\rm \frac1y-\frac1x\)
Answer» Correct Answer - Option 3 : \(\rm \frac1x-\frac1y\) Concept: Trigonometric Identities: \(\rm \cot\theta=\frac{1}{\tan\theta}\). \(\rm \cot(A\pm B)=\frac{\mp1+cot A\cot B}{\pm\cot A+\cot B}\). Calculation: Given: tan A - tan B = x ... (1) And, cot A - cot B = y ... (2) ⇒ \(\rm \frac{1}{\tan A}-\frac{1}{\tan B}=y\) ⇒ \(\rm \frac{\tan B-\tan A}{\tan A\tan B}=y\) ⇒ \(\rm \frac{-x}{\tan A\tan B}=y\) ... [Using (1).] ⇒ \(\rm \cot A\cot B=-\frac{y}{x}\) ... (3) Now, \(\rm \cot(A- B)=\frac{+1+cot A\cot B}{-\cot A+\cot B}\) = \(\rm \frac{+1-\frac yx}{-y}\) ... [Using (2) and (3).] = \(\rm \frac{y-x}{xy}\) = \(\rm \frac{1}{x}-\frac{1}{y}\).

If tan A - tan B = x and cot A - cot B = y, then cot (A - B) = ?1. x - y2. \(\rm \frac1x+\frac1y\)3. \(\rm \frac1x-\frac1y\)4. \(\rm \frac1y-\frac1x\)

Answer» Correct Answer - Option 3 : \(\rm \frac1x-\frac1y\)

Concept:

Trigonometric Identities:

Calculation:

Given: tan A - tan B = x ... (1)

And, cot A - cot B = y ... (2)

⇒ \(\rm \frac{1}{\tan A}-\frac{1}{\tan B}=y\)

⇒ \(\rm \frac{\tan B-\tan A}{\tan A\tan B}=y\)

⇒ \(\rm \frac{-x}{\tan A\tan B}=y\) ... [Using (1).]

⇒ \(\rm \cot A\cot B=-\frac{y}{x}\) ... (3)

Now, \(\rm \cot(A- B)=\frac{+1+cot A\cot B}{-\cot A+\cot B}\)

= \(\rm \frac{+1-\frac yx}{-y}\) ... [Using (2) and (3).]

= \(\rm \frac{y-x}{xy}\)

= \(\rm \frac{1}{x}-\frac{1}{y}\).