If $\int \frac{d\theta }{{\cos }^2\theta \left(\tan 2\theta +\sec 2\theta \right)}=$ $\lambda \tan \theta +2{\log }_e\left|f\left(\theta \right)\right|+C$ where $C$ is a constant of integration, then the ordered pair $\left(\lambda ,f\left(\theta \right)\right)$ is equal to:

• A

  $\left(1,1-\tan \theta \right)$
• B

  $\left(-1,1-\tan \theta \right)$
• Correct answer
  C

  $\left(-1,1+\tan \theta \right)$
• D

  $\left(1,1+\tan \theta \right)$

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