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 5 of 5: Apply supplementary-angle identities and conclude
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cos⁡2π7−cos⁡3π7−cos⁡π7=−12⟹cos⁡π7−cos⁡2π7+cos⁡3π7=12\cos\frac{2\pi}{7}-\cos\frac{3\pi}{7}-\cos\frac{\pi}{7}=-\frac12\Longrightarrow\boxed{\cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7}=\frac12}
Detailed analysis

Use cos⁡(π−x)=−cos⁡x\cos(\pi-x)=-\cos x: cos⁡(4π/7)=−cos⁡(3π/7)\cos(4\pi/7)=-\cos(3\pi/7) and cos⁡(6π/7)=−cos⁡(π/7)\cos(6\pi/7)=-\cos(\pi/7). Substituting these into the three-cosine identity and multiplying by −1-1 gives the required result.