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 3 of 5: Write the cosine sum explicitly
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cos⁡2π7+cos⁡4π7+cos⁡6π7+cos⁡8π7+cos⁡10π7+cos⁡12π7=−1\cos\frac{2\pi}{7}+\cos\frac{4\pi}{7}+\cos\frac{6\pi}{7}+\cos\frac{8\pi}{7}+\cos\frac{10\pi}{7}+\cos\frac{12\pi}{7}=-1
Detailed analysis

The real-part equation is exactly the displayed sum, since ωi=cis⁡(2πi/7)\omega^i=\operatorname{cis}(2\pi i/7). Pairing ii with 7−i7-i changes the six terms into two copies of the first three.