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 π/17.

Step 2: Key Formula or Approach (Alternate Method):
Apply product-to-sum formula on the last term, then use the property cos(π - θ) = -cos θ to cancel terms.

Step 3: Detailed Explanation:
Let E = cos(5π/17) + cos(7π/17) + 2cos(11π/17)cos(π/17). Apply product-to-sum: 2cos(11π/17)cos(π/17) = cos(12π/17) + cos(10π/17). Now E = cos(5π/17) + cos(7π/17) + cos(12π/17) + cos(10π/17). Note: 12π/17 = π - 5π/17 → cos(12π/17) = -cos(5π/17). And 10π/17 = π - 7π/17 → cos(10π/17) = -cos(7π/17). Substitute: E = cos(5π/17) + cos(7π/17) - cos(5π/17) - cos(7π/17) = 0.

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