If $\tan \left(\frac{\pi}{4}+\alpha\right)=\tan ^3\left(\frac{\pi}{4}+\beta\right)$, then $\tan (\alpha+\beta) \cot (\alpha-\beta)=$

• A

  $\sec ^2 2 \beta+\tan ^2 2 \beta$
• B

  $\operatorname{cosec}^2 2 \beta+\cot ^2 2 \beta$
• Correct answer
  C

  $2\left(\sec ^2 2 \beta+\tan ^2 2 \beta\right)$
• D

  $4\left(\sec ^2 2 \beta+\tan ^2 2 \beta\right)$

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