Question 5 - A-Level Maths

Given that $4 \tan x + 3 \cos x + \frac { 1 } { \cos x } = 0$, show, without using a calculator, that $\sin x = - \frac { 2 } { 3 }$.
Hence, showing all necessary working, solve the equation $$4 \tan \left( 2 x - 20 ^ { \circ } \right) + 3 \cos \left( 2 x - 20 ^ { \circ } \right) + \frac { 1 } { \cos \left( 2 x - 20 ^ { \circ } \right) } = 0$$ for $0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.