Question

General value of $\theta$ satisfying the equation $\tan^{2}\theta + \sec 2\theta = 1$is

• A. $m\pi,n\pi + \frac{\pi}{3}$
• B. $m\pi,n\pi \pm \frac{\pi}{3}$
• C. $m\pi,n\pi \pm \frac{\pi}{6}$
• D. None of these

Answer 1

Answer: (B)

$m\pi,n\pi \pm \frac{\pi}{3}$

Answer 2

$\tan^{2}\theta + \sec 2\theta = 1$ ⇒ $\tan^{2}\theta + \frac{1 + \tan^{2}\theta}{1 - \tan^{2}\theta} = 1$

⇒ $\tan ^ { 2 } \theta - \tan ^ { 4 } \theta + 1 + \tan ^ { 2 } \theta = 1 - \tan ^ { 2 } \theta$ $\tan^{4}\theta - 3\tan^{2}\theta = 0$ ⇒ $\tan^{2}\theta(\tan^{2}\theta - 3) = 0$ ⇒ $\tan^{2}\theta = 0$ and $\tan^{2}\theta = 3$

$\tan^{2}\theta = \tan^{2}0$ and $\tan^{2}\theta = \tan^{2}\frac{\pi}{3}$ ⇒ $\theta = m\pi$ and

$\theta = n\pi \pm \frac{\pi}{3}.$