Step 1: Understanding the Question:
The goal is to simplify a given expression involving side lengths of a triangle $ABC$ with a known angle $\angle A = 30^\circ$. Step 2: Key Formula or Approach:
We use the Cosine Rule: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$. Step 3: Detailed Explanation:
The expression is:
\[ E = (1 + \frac{a}{c} + \frac{b}{c})(1 + \frac{c}{b} - \frac{a}{b}) \]
\[ E = (\frac{c + a + b}{c})(\frac{b + c - a}{b}) \]
\[ E = \frac{(b + c + a)(b + c - a)}{bc} \]
\[ E = \frac{(b + c)^2 - a^2}{bc} \]
\[ E = \frac{b^2 + c^2 + 2bc - a^2}{bc} \]
From the Cosine Rule, $b^2 + c^2 - a^2 = 2bc \cos A$.
Substituting this into the expression:
\[ E = \frac{2bc \cos A + 2bc}{bc} = 2(\cos A + 1) \]
Given $\angle A = 30^\circ$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$.
\[ E = 2(\frac{\sqrt{3}}{2} + 1) = \sqrt{3} + 2 \] Step 4: Final Answer:
The value is $\sqrt{3} + 2$.