Question

Find all values of $\theta$ in the interval $(-\frac{\pi }{2},\frac{\pi }{2})$ satisfying the equation $(1-\tan \theta )(1+\tan \theta ){\sec }^2\theta +2^{{\tan }^2\theta }=0$. (IIT-JEE, 1996)

Answer 1

Given that $(1-\tan \theta )(1+\tan \theta ){\sec }^2\theta +2^{{\tan }^2\theta }=0$

$\Rightarrow (1-\tan \theta )(1+\tan \theta )+2^{{\tan }^2\theta }=0$

Let us put ${\tan }^2\theta =t$

$\therefore (1-t)(1+t)+2^t=0$

It is clearly satisfy by $t=3$

as $-8+8=0$, we get $ta{n}^2\theta =3$

$\therefore \theta =\pm \pi /3$ in the given interval.