Step 1: Understanding the Question:
We are given a condition involving the sides of a triangle and need to find the value of a trigonometric expression involving its angle $A$.
Step 2: Key Formula or Approach:
- Triple angle formula: $\cos 3A = 4\cos^3 A - 3\cos A$
- Cosine rule: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$
Step 3: Detailed Explanation:
First, simplify the given expression:
\[ E = \frac{\cos 3A}{\cos A} + 2 \]
\[ E = \frac{4\cos^3 A - 3\cos A}{\cos A} + 2 \]
\[ E = 4\cos^2 A - 3 + 2 = 4\cos^2 A - 1 \]
Now, use the given condition $2a^2 = b^2 + c^2 \implies b^2 + c^2 = 2a^2$.
Substitute this into the cosine rule for $\cos A$:
\[ \cos A = \frac{(b^2 + c^2) - a^2}{2bc} = \frac{2a^2 - a^2}{2bc} = \frac{a^2}{2bc} \]
To find a constant value, let's consider a specific case satisfying $2a^2 = b^2 + c^2$.
Let $a = b = c = 1$ (Equilateral triangle case).
This satisfies $2(1)^2 = (1)^2 + (1)^2$, which is $2 = 2$.
In an equilateral triangle, $A = 60^\circ$.
Substitute $A = 60^\circ$ into the simplified expression:
\[ E = 4\cos^2(60^\circ) - 1 \]
\[ E = 4\left(\frac{1}{2}\right)^2 - 1 = 4\left(\frac{1}{4}\right) - 1 = 1 - 1 = 0 \]
Step 4: Final Answer:
The value of the expression is $0$.