Question

The general solution of the equation ${\tan }^4\theta -1=2^{{\tan }^2\theta }$ is

• A. $2n\pi +\frac{\pi }{3}$
• B. $2n\pi \pm \frac{\pi }{3}$
• C. $n\pi +(-1{)}^n\frac{2\pi }{3}$
• D. None of these

Answer 1

The correct option is B $2n\pi \pm \frac{\pi }{3}$

$(1-{\tan }^4\theta )+2^{{\tan }^2\theta }=0$
Let ${\tan }^2\theta =t$.
Hence
$1-t^2+2^t=0$
Or
$t^2-1=2^t$.
From inspection of graph we get
$t^2=9$ is one solution.
Or
$t=3$ since ${\tan }^2\theta =-3$ is not possible.
Or
${\tan }^2\theta =3$
Or
$\tan \theta =\pm \sqrt{3}$.
Or
$\theta =\pm \frac{\pi }{3}$.
Hence
$\theta =2n\pi \pm \frac{\pi }{3}$.