Pls solve this question for me

Dear Student.

$$\begin{gathered} 1) 1+\tan 2\theta \tan \theta =sec 2\theta \\ 1+\frac{\sin 2\theta }{\cos 2\theta }\times \frac{\sin \theta }{\cos \theta }=1+\frac{2\sin \theta \cos \theta }{\cos 2\theta }\times \frac{\sin \theta }{\cos \theta } \\ =1+\frac{2{\sin }^2\theta }{\cos 2\theta }=\frac{\cos 2\theta +2{\sin }^2\theta }{\cos 2\theta } \\ Since \cos 2\theta =1-2{\sin }^2\theta \\ =\frac{1-2{\sin }^2\theta +2{\sin }^2\theta }{\cos 2\theta } \\ =\frac{1}{\cos 2\theta }=sec 2\theta \end{gathered}$$

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