Number of distinct solutions of $\sec \theta +\tan \theta =\sqrt{3},0\le \theta \le 3\pi$ , is

$\frac{1+\sin \theta }{\cos \theta }=\sqrt{3}$

$\frac{{\sin }^2{\displaystyle \frac{\theta }{2}}+{\cos }^2{\displaystyle \frac{\theta }{2}}+2\sin {\displaystyle \frac{\theta }{2}}\cos {\displaystyle \frac{\theta }{2}}}{{\cos }^2{\displaystyle \frac{\theta }{2}}-{\sin }^2{\displaystyle \frac{\theta }{2}}}=\sqrt{3}$

$\frac{{\left(\cos {\displaystyle \frac{\theta }{2}}+\sin {\displaystyle \frac{\theta }{2}}\right)}^2}{\left(\cos {\displaystyle \frac{\theta }{2}}-\sin {\displaystyle \frac{\theta }{2}}\right)\left(\cos \frac{\theta }{2}+\sin \frac{\theta }{2}\right)}=\sqrt{3}$

$\frac{\left(\cos {\displaystyle \frac{\theta }{2}}+\sin {\displaystyle \frac{\theta }{2}}\right)}{\left(\cos \frac{\theta }{2}-\sin \frac{\theta }{2}\right)}=\sqrt{3}$

$\frac{\left(1+tan{\displaystyle \frac{\theta }{2}}\right)}{\left(1-tan\frac{\theta }{2}\right)}=\sqrt{3}$

$\tan \left(\frac{\pi }{4}+\frac{\theta }{2}\right)=\sqrt{3}$

g sol $\frac{\pi }{4}+\frac{\theta }{2}=n\pi +\frac{\pi }{3}$

$\theta =2n\pi +\frac{2\pi }{3}-\frac{\pi }{2}$

n=0 $\theta =\frac{\pi }{6}$

n=1 $\theta =2\pi +\frac{\pi }{6}$ two values