Question

Find all values of $\theta$ in the interval $(-\pi /2,\pi /2)$ satisfying the equation $(1-tan\theta )(1+tan\theta )se{c}^2\theta +2^{ta{n}^2\theta }=0$

• A. $\pm \frac{\pi }{3}$
• B. $\pm \frac{\pi }{4}$
• C. $\pm \frac{\pi }{6}$
• D. $\pm \frac{\pi }{8}$

Answer 1

The correct option is A $\pm \frac{\pi }{3}$

$(1-tan\theta )(1+tan\theta )se{c}^2\theta +2^{ta{n}^2\theta }=0\Rightarrow (1-{\tan }^2\theta )(1+{\tan }^2\theta )+2^{ta{n}^2\theta }=0\Rightarrow 1-{\tan }^4\theta +2^{ta{n}^2\theta }=0$

Substitute ${\tan }^2\theta =x$

$\therefore 1-x^2+2^x=0\Rightarrow x^2-1=2^x$

Only $x=3$ satisfy this

${\tan }^2\theta =3\Rightarrow \tan \theta =\pm \sqrt{3}\Rightarrow \theta =\pm \frac{\pi }{3}$