Find the general solution for ^2 2 x=1- 2 x equation. - Vidyadip

Question

Find the general solution for $\sec ^2 2 x=1-\tan 2 x$ equation.

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Answer

$\sec ^2 2 x=1-\tan 2 x$
$ \Rightarrow 1 + {\tan ^2}2x = 1 - \tan 2x$
$ \Rightarrow {\tan ^2}2x + \tan 2x = 0$
$ \Rightarrow \tan 2x(\tan 2x + 1) = 0$
$ \Rightarrow $ either tan 2x = 0 or tan 2x + 1 = 0
$ \Rightarrow 2x = n\pi $ or $\tan 2x = - 1 = - \tan \frac{\pi }{4} = \tan \left( { - \frac{\pi }{4}} \right),n \in Z$
$ \Rightarrow x = \frac{{n\pi }}{2}$ or $x = \frac{{n\pi }}{2} + \frac{{3\pi }}{8},n \in Z$

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