If \( \Large \tan A = 2 \tan B + \cot B,\ then\ 2 \tan \left(A-B\right) \) is equal to:

A) \( \Large \tan B \)

B) \( \Large 2 \tan B \)

C) \( \Large \cot B \)

D) \( \Large 2 \cot B \)

Correct answer:
C) \( \Large \cot B \)

Description for Correct answer:

Given that

\( \Large \tan A = 2 \tan B + \cot B \) ...(i)

Now, \( \Large 2 \tan \left(A-B\right)=2 \left(\frac{\tan A - \tan B}{1+ \tan A \tan B}\right) \)

= \( \Large 2 \frac{ \left(2 \tan B + \cot B - \tan B\right) }{1+ \left(2 \tan B + \cot B\right) \tan B } \) From (i)

= \( \Large 2 \frac{\tan B + \cot B}{2 \left(1+\tan^{2}B\right) } = \frac{\cot B \left( \tan^{2}B+1 \right) }{ \left(1+ \tan^{2}B\right) } = \cot B \)



Please provide the error details in above question