If $\tan \theta +3\cot \theta =5\sec \theta$ , then $\theta =\left(n\in z\right)$  

$\frac{\sin \theta }{\cos \theta }+3.\frac{\cos \theta }{\sin \theta }=\frac{5}{\cos \theta }$

${\sin }^2\theta +3{\cos }^2\theta =5\sin \theta$

$2{\sin }^2+5\sin \theta -3=0$

$\left(2\sin \theta -1\right)\left(\sin \theta +3\right)=0$

$\sin \theta =-3$  is not possible

$\sin \theta =\frac{1}{2}$  

$\theta =\frac{\pi }{6}$

$\therefore$  G.S is $\theta =n\pi +{\left(-1\right)}^n\frac{\pi }{6}$