What is the value of \cos(\frac{5\pi}{17})+\cos(\frac{7\pi}{17})+2\cos(\frac{11\pi}{17})\cos(\frac{\pi}{17})?
- A. 0 ✓
- B. 1
- C. 4\cos(\frac{6\pi}{17})\cos(\frac{\pi}{17})
- D. 4\cos(\frac{11\pi}{17})\cos(\frac{\pi}{17})
Correct Answer: A. 0
Explanation
Applying the sum-to-product formula on the first two terms: \cos(\frac{5\pi}{17}) + \cos(\frac{7\pi}{17}) = 2\cos(\frac{6\pi}{17})\cos(\frac{\pi}{17}). Factoring out 2\cos(\frac{\pi}{17}) from the entire expression gives 2\cos(\frac{\pi}{17})[\cos(\frac{6\pi}{17}) + \cos(\frac{11\pi}{17})]. Since \frac{6\pi}{17} + \frac{11\pi}{17} = \pi, we have \cos(\frac{11\pi}{17}) = -\cos(\frac{6\pi}{17}). Thus, the term inside the brackets is zero, making the overall sum 0.