Question

Solve $\tan \theta +3\cot \theta =5\sec \theta$

Answer 1

$\tan \theta +3\cot \theta =5\sec \theta$

$\frac{\sin \theta }{\cos \theta }+3\frac{\cos \theta }{\sin \theta }=5\frac{1}{\cos \theta }$

$\frac{\sin \theta \cos \theta +3\cos \theta \sin \theta }{\sin \theta \cos \theta }=5\frac{1}{\cos \theta }$

$\frac{4\sin \theta \cos \theta }{\sin \theta }=5$

$4\sin \theta \cos \theta =5\sin \theta$

$\sin \theta (4\cos \theta -5)=0$

So either $\sin \theta =0$ or $4\cos \theta -5=0$

So either

$\theta =n\pi$ or $\cos \theta =\frac{5}{4}$ which is not possible because $\cos \theta \le 1$

So

$\theta =n\pi$