Question

Let $S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$.
Then the number of elements in the set
$A =\{\theta \in S : \tan \theta(1+\sqrt{5} \tan (2 \theta))=\sqrt{5}-\tan (2 \theta)\}$

Answer 1

$ \tan \theta+\sqrt{5} \tan 2 \theta \tan \theta=\sqrt{5}-\tan 2 \theta$
$ \tan 3 \theta=\sqrt{5}$
$\theta=\frac{ n \pi}{3}+\frac{\alpha}{3} ; \quad \tan \alpha=\sqrt{5}$
Five solution