$$\frac { \cot x } { \tan x + \cot x } = 4 \sin x - 3$$ Given this, what is the value of $\sin x$?\\ A) $3 - 2 \sqrt { 2 }$\\ B) $1 - \sqrt { 3 }$\\ C) $- 1 + \sqrt { 2 }$\\ D) $- 1 + \sqrt { 3 }$\\ E) $- 2 + 2 \sqrt { 2 }$