(i) Show that

\(\frac{\tan x}{1+\operatorname{sec}x}+\frac{1+\operatorname{sec}x}{\tan x}\equiv\frac{2}{\sin x}.\)

(ii) Hence solve

\(\frac{\tan x}{1+\operatorname{sec}x}+\frac{1+\operatorname{sec}x}{\tan x}=1+3\sin x\)

for \(0^\circ\lt x\lt180^\circ\).