2.11: Cubic in sin from sec–tan equation

Show that $13\tan x \sec x - 4\sin x - 5\sec^{2} x = 0$ can be written as $4\sin^{3} x + 9\sin x - 5 = 0$.

Using your answers to part (a)(ii) and part (b)(i), solve the equation $13\tan x \sec x - 4\sin x - 5\sec^{2} x = 0$ for $0 < x < 2\pi$ radians.