Question

Find the general solution of the equation ${\sec }^{2}2x=1-\tan 2x$

Answer 1

Step 1: Simplification
Given: ${\sec }^{2}2x=1-\tan 2x$
$\Rightarrow 1+{\tan }^{2}x=1-\tan 2x$
$\Rightarrow {\tan }^{2}2x+\tan 2x=0$
$\Rightarrow \tan 2x(\tan 2x+1)=0$
$\Rightarrow \tan 2x=0$ or $\tan 2x+1=0$
$\Rightarrow \tan 2x=0$ or $\tan 2x=-1$

We find the general solution for both equations separately.
Step 2:
General solution for $\tan 2x=0$
$\tan 2x=0$
$\Rightarrow \tan 2x=\tan 0$
General solution is $2x=n\pi +0$
$\Rightarrow x=\frac{n\pi }{2}$

Step $3:$
General solution for $\tan 2x=-1$
$\tan 2x=-1$
$\Rightarrow \tan 2x=\tan \frac{3\pi }{4}$
General solution is $2x=n\pi +\frac{3\pi }{4},n\in Z$
$x=\frac{n\pi }{2}+\frac{3\pi }{8},$ where $n\in Z$