Question

Solve $(1-{\tan }^2\theta )(1+{\tan }^2\theta )+2^{{\tan }^2\theta }=0$ in the interval $(-\pi /2,\pi /2)$.

Answer 1

Let us put ${\tan }^2\theta =t$
$\therefore (1-t)(1+t)+2^t=0$
or $1-t^2+2^t=0$
It is clearly satisfied by $t=3$
as $-8+8=0$
$\therefore {\tan }^2\theta =3$ or $\tan \theta =\pm \sqrt{3}$
$\therefore \theta =\pm \pi /3$ in the given interval.