(a)(i) Use the factor theorem to show that \(2x-1\) is a factor of \(p(x)\), where \(p(x)=4x^3+9x-5\).

(ii) Write \(p(x)\) as a product of linear and quadratic factors.

(b)(i) Show that

\(13\tan x\operatorname{sec}x-4\sin x-5\operatorname{sec}^2x=0\)

can be written as

\(4\sin^3x+9\sin x-5=0.\)

(ii) Using your answers to part (a)(ii) and part (b)(i), solve

\(13\tan x\operatorname{sec}x-4\sin x-5\operatorname{sec}^2x=0\)

for \(0\lt x\lt 2\pi\) radians.