Question

The general solution of $4{\tan }^2\theta =3{\sec }^2\theta$ is $\theta =n\pi \pm \frac{\pi }{m}$. Then, find the value of $m$.

Answer 1

$\because 4{\tan }^2\theta =3{\sec }^2\theta$
$\frac{4{\sin }^2\theta }{{\cos }^2\theta }=\frac{3}{{\cos }^2\theta }$
${\cos }^2\theta \ne 0\therefore \theta \ne (2n+1)\frac{\pi }{2},n\in I$
Hence, $4{\sin }^2\theta =3$
$\Rightarrow {\sin }^2\theta ={\left(\frac{\sqrt{3}}{2}\right)}^2$
$\Rightarrow {\sin }^2\theta ={\sin }^2\frac{\pi }{3}$
$\Rightarrow \theta =n\pi \pm \frac{\pi }{3},n\in I$