Prove the following


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

  1. $(-1,1+\tan \theta)$

  2. $(-1,1-\tan \theta)$

  3. $(1,1-\tan \theta)$

  4. $(1,1+\tan \theta)$

Correct Option: 1


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