Question

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

Answer 1

Hint - Express ${\sec ^2}\theta = 1 + {\tan ^2}\theta$ and solve the problem.

Complete step-by-step answer:

The given equation is ${\sec ^2}2x$
So, if we express in terms of tan , it would be equal to
$1 + {\tan ^2}2x$ =1-tan2x
On shifting and rearranging the terms, we get this to be equal to
tan2x(tan2x+1)=0
So, from this we get the value of tan2x=0 or tan2x=-1
Shifting tan to the other side, we get
$\Rightarrow 2x = {\tan ^{ - 1}}0$ or $2x = {\tan ^{ - 1}}( - 1)$
$\Rightarrow 2x = n\pi ,n\pi - \dfrac{\pi }{4}$
So, from this we get the general solution of the equation, that is
$x = \dfrac{{n\pi }}{2},\dfrac{{n\pi }}{2} - \dfrac{\pi }{8}$
So, this is the general solution of the equation.

Note: In accordance to the value which has to be found , make use of the appropriate trigonometric identities and solve this type of problem and also express the equation given in a convenient form before solving it further.