(i) Start with the equation:

\(4 \tan x + 3 \cos x + \frac{1}{\cos x} = 0\)

Multiply through by \(\cos x\) to eliminate the fraction:

\(4 \sin x + 3 \cos^2 x + 1 = 0\)

Use the identity \(\cos^2 x = 1 - \sin^2 x\):

\(4 \sin x + 3(1 - \sin^2 x) + 1 = 0\)

Simplify to form a quadratic in \(\sin x\):

\(3 \sin^2 x - 4 \sin x - 4 = 0\)

Solving this quadratic gives \(\sin x = -\frac{2}{3}\).

(ii) Substitute \(\sin(2x - 20^\circ) = -\frac{2}{3}\) into the equation:

\(2x - 20^\circ = 221.8^\circ, 318.2^\circ\)

Solve for \(x\):

\(x = 120.9^\circ, 169.1^\circ\)