(i) Given that \(4 \tan x + 3 \cos x + \frac{1}{\cos x} = 0\), show, without using a calculator, that \(\sin x = -\frac{2}{3}\).

(ii) Hence, showing all necessary working, solve the equation \(4 \tan(2x - 20^\circ) + 3 \cos(2x - 20^\circ) + \frac{1}{\cos(2x - 20^\circ)} = 0\) for \(0^\circ \leq x \leq 180^\circ\).