Question

Find the general value of $\theta $ if $2{\tan ^2}\theta = {\sec ^2}\theta $

Answer 1

Hint- In order to solve this question we will use the trigonometric functions and then by applying the conditions given in the question we can easily solve this question.
General solution for $\sin {\text{x}} = 0$ will be, $x = n\pi, $ where $n \in I$( $I$ = integer) . Similarly, general solution for $\cos x$ will be $x = \dfrac{{\left( {2n + 1} \right)\pi }}{2},n \in I,$ ($I$ = integer ) as cos x = 0 has value equal to 0 at $\dfrac{\pi }{2},\dfrac{{3\pi }}{2},\dfrac{{5\pi }}{2},\dfrac{{ - 7\pi }}{2}, \dfrac{{ - 11\pi }}{2}$ etc.

Complete step-by-step answer:
It is given that,
$2{\tan ^2}\theta = {\sec ^2}\theta $
$\dfrac{{2{{\sin }^2}\theta }}{{{{\cos }^2}\theta }} = \dfrac{1}{{{{\cos }^2}\theta }}$
Now, we know that $\cos \theta \ne 0$ because if $\cos \theta = 0$ then the RHS of the equation doesn’t exist.
$2{\sin ^2}\theta = 1$
${\sin ^2}\theta = \dfrac{1}{2}$
$ \Rightarrow \sin \theta = \pm \dfrac{1}{{\sqrt 2 }}$ [$\because \sin(\dfrac{\pi}{4}) = \dfrac{1}{{\sqrt 2 }}$]
$ \Rightarrow \theta = n\pi \pm \dfrac{\pi }{4};$$n\in I$(where $n$ belongs to $I$ ), is the correct answer.

Note- Whenever we come up with these types of problems, we should know that we have to remember the general solutions for trigonometric variables. Also we have to not be confused in calculations and always remember the signs we have to use. By these basics one can solve these types of questions.