Find the general solution of ^2 2 =1- 2 - Vidyadip

Question

Find the general solution of $\sec ^2 2 \theta=1-\tan 2 \theta$

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Answer

Given equation is $\sec ^2 2 \theta=1-\tan 2 \theta$
$ \therefore \quad 1+\tan ^2 2 \theta=1-\tan 2 \theta$
$\therefore \quad \tan ^2 2 \theta+\tan 2 \theta=0$
$\therefore \quad \tan 2 \theta(\tan 2 \theta+1)=0$
$\therefore \quad \tan 2 \theta=0 \text { or } \tan 2 \theta+1=0$
$\therefore \quad \tan 2 \theta=\tan 0 \text { or } \tan 2 \theta=\tan \frac{3 \pi}{4}$
$\therefore \quad 2 \theta= n \pi \text { or } 2 \theta= n \pi+\frac{3 \pi}{4}, \text { where } n \in Z .$
$\therefore \quad \theta=\frac{n \pi}{2} \text { or } \theta=\frac{n \pi}{2}+\frac{3 \pi}{8}, \text { where } n \in Z \text { is the required general solution. } $

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