MathLabs

Problem 5

Prove that cos⁡π7−cos⁡2π7+cos⁡3π7=12\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}=\frac12.
Step 2 of 5: Take real parts
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∑i=16ωi=−1⟹∑i=16Re⁡(ωi)=−1\sum_{i=1}^6\omega^i=-1\Longrightarrow\sum_{i=1}^6\operatorname{Re}(\omega^i)=-1
Detailed analysis

Removing the root 11 gives ∑i=16ωi=−1\sum_{i=1}^6\omega^i=-1. Taking real parts, and using Re⁡(cis⁡α)=cos⁡α\operatorname{Re}(\operatorname{cis}\alpha)=\cos\alpha, gives the corresponding cosine sum equal to −1-1.