If \( \int \frac{\mathrm{d} \theta}{\cos ^{2} \theta(\tan 2 \theta+\sec 2 \theta)}=\lambda \tan \theta+2 \log _{\mathrm{…

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

[JEE Main 2020, 9 Jan (Shift 2)]