Question

If $\tan A=2\tan B+\cot B$, then $2\tan (A-B)$ equals

• A. $\tan B$
• B. $2\tan B$
• C. $\cot B$
• D. $2\cot B$

Answer 1

The correct option is C $\cot B$

Given $\tan A=2\tan B+\cot B$

$⟹\tan A\tan B=2{\tan }^{2}B+1$

$\tan A-\tan B=2\tan B+\cot B-\tan B=\tan B+\cot B=\frac{{\tan }^{2}B+1}{\tan B}$

$2\tan (A-B)=\frac{2(\tan A-\tan B)}{1+\tan A\tan B}$

$=\frac{2(\frac{1+{\tan }^{2}B}{\tan B})}{1+2{\tan }^{2}B+1}$

$=\frac{2(1+{\tan }^{2}B)}{2(1+{\tan }^{2}B)\tan B}$

$=\cot B$