Step 1: Understanding the Concept:
This problem requires simplifying a trigonometric expression using product-to-sum and other trigonometric identities. The goal is to see if terms cancel out.
Step 2: Key Formula or Approach:
We will use the product-to-sum identity:
\[ 2 \cos A \cos B = \cos(A+B) + \cos(A-B) \]
We will also use the identity $\cos(\pi - \theta) = -\cos\theta$.
Step 3: Detailed Explanation:
The given expression is:
\[ \cos\frac{5\pi}{17} + \cos\frac{7\pi}{17} + 2\cos\frac{11\pi}{17}\cos\frac{\pi}{17} \]
First, let's simplify the last part of the expression using the product-to-sum formula with $A=\frac{11\pi}{17}$ and $B=\frac{\pi}{17}$:
\[ 2\cos\frac{11\pi}{17}\cos\frac{\pi}{17} = \cos\left(\frac{11\pi}{17}+\frac{\pi}{17}\right) + \cos\left(\frac{11\pi}{17}-\frac{\pi}{17}\right) \]
\[ = \cos\left(\frac{12\pi}{17}\right) + \cos\left(\frac{10\pi}{17}\right) \]
Now substitute this back into the original expression:
\[ \text{Expression} = \cos\frac{5\pi}{17} + \cos\frac{7\pi}{17} + \cos\frac{12\pi}{17} + \cos\frac{10\pi}{17} \]
Next, we use the identity $\cos(\pi - \theta) = -\cos\theta$ to relate the terms.
\[ \cos\frac{12\pi}{17} = \cos\left(\pi - \frac{5\pi}{17}\right) = -\cos\frac{5\pi}{17} \]
\[ \cos\frac{10\pi}{17} = \cos\left(\pi - \frac{7\pi}{17}\right) = -\cos\frac{7\pi}{17} \]
Substitute these simplified forms back into the expression:
\[ \text{Expression} = \cos\frac{5\pi}{17} + \cos\frac{7\pi}{17} - \cos\frac{5\pi}{17} - \cos\frac{7\pi}{17} \]
All the terms cancel each other out.
\[ \text{Expression} = 0 \]
Step 4: Final Answer:
The value of the expression is 0. Therefore, option (A) is correct.