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1-\cos^2x-\cos^2y- y-\cos^2z +2 \cos x \cos y \cos z = ...

Source: Polish MO Second Round 1961 p3

August 31, 2024
trigonometry

Problem Statement

Prove that for any angles x,y,zx,y,z holds the equality 1−cos⁡2x−cos⁡2y−y−cos⁡2z+2cos⁡xcos⁡ycos⁡z=4sin⁡x+y+z2sin⁡x+y−z2sin⁡x−y+z2sin⁡−x−y+z2.1-\cos^2x-\cos^2y- y-\cos^2z +2 \cos x \cos y \cos z= 4 \sin \frac{x+y+z}{2} \sin \frac{x+y-z}{2} \sin \frac{x-y+z}{2} \sin\frac{-x-y+z}{2}.
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