Question

The value of \(\cos\frac{5\pi}{17} + \cos\frac{7\pi}{17} + 2\cos\frac{11\pi}{17}\cos\frac{\pi}{17}\) is

• A. 0
• B. 1
• C. -1
• D. 1/2

Hint:

When dealing with trigonometric sums involving fractions of \(\pi\), always look for pairs of angles that add up to \(\pi\) or \(\pi/2\). This allows you to use identities like \(\cos(\pi - \theta) = -\cos(\theta)\) or \(\cos(\pi/2 - \theta) = \sin(\theta)\) to simplify the expression. Here, recognizing \(12\pi/17\) as \(\pi - 5\pi/17\) was the key to solving the problem.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to evaluate a trigonometric expression involving cosine functions with arguments that are multiples of \(\frac{\pi}{17}\).

Step 2: Key Formula or Approach:
We will use the product-to-sum formula:
\[ 2 \cos A \cos B = \cos(A+B) + \cos(A-B) \]
And the property \(\cos(\pi - \theta) = -\cos(\theta)\).

Step 3: Detailed Explanation:
Let the given expression be \(E\).
\[ E = \cos\frac{5\pi}{17} + \cos\frac{7\pi}{17} + 2\cos\frac{11\pi}{17}\cos\frac{\pi}{17} \]
First, apply the product-to-sum formula to the third term:
\[ 2\cos\frac{11\pi}{17}\cos\frac{\pi}{17} = \cos\left(\frac{11\pi}{17} + \frac{\pi}{17}\right) + \cos\left(\frac{11\pi}{17} - \frac{\pi}{17}\right) \]
\[ = \cos\left(\frac{12\pi}{17}\right) + \cos\left(\frac{10\pi}{17}\right) \]
Now substitute this back into the expression for \(E\):
\[ E = \cos\frac{5\pi}{17} + \cos\frac{7\pi}{17} + \cos\frac{12\pi}{17} + \cos\frac{10\pi}{17} \]
We can rewrite the arguments of the last two terms using \(\pi\).
Notice that \(\frac{12\pi}{17} = \pi - \frac{5\pi}{17}\).
And \(\frac{10\pi}{17} = \pi - \frac{7\pi}{17}\).
Let's apply the identity \(\cos(\pi - \theta) = -\cos(\theta)\).
\[ \cos\left(\frac{12\pi}{17}\right) = \cos\left(\pi - \frac{5\pi}{17}\right) = -\cos\left(\frac{5\pi}{17}\right) \]
\[ \cos\left(\frac{10\pi}{17}\right) = \cos\left(\pi - \frac{7\pi}{17}\right) = -\cos\left(\frac{7\pi}{17}\right) \]
Now substitute these simplified terms back into the expression for \(E\):
\[ E = \cos\frac{5\pi}{17} + \cos\frac{7\pi}{17} - \cos\frac{5\pi}{17} - \cos\frac{7\pi}{17} \]
All the terms cancel each other out.
\[ E = 0 \]

Step 4: Final Answer:
The value of the expression is 0.