Geometric Proof of the Double Angle Cosine Formula

This GeoGebra example demonstrates the geometric proof of the double angle cosine formula cos(2α) = cos²α - sin²α. Using geometric constructions including circles, rays, angles, perpendicular lines, and reflections, it visually illustrates the trigonometric relationship between double angles and single angles. Through precise constructions of intersections, segments, and angles, it helps learners from a geometric perspective understand the essence of the double angle cosine formula. The interactive applet allows students to manipulate the angle parameter and observe in real time how the corresponding side lengths and areas transform, reinforcing the connection between algebraic identities and geometric reasoning. Key construction elements include a unit circle centered at the origin, a ray forming the angle α with the horizontal axis, the reflected ray representing the angle 2α, perpendicular projections onto coordinate axes, and carefully marked segment lengths that correspond to sin α, cos α, sin 2α, and cos 2α. By comparing the length of the horizontal projection associated with 2α against the combined lengths derived from the projections related to α, the proof unfolds step by step before the learner's eyes. This visual approach makes an otherwise abstract identity intuitive and memorable. The applet is particularly suited for high school trigonometry instruction, serving both as a classroom demonstration tool and as a resource for independent study. Teachers can use it to guide students through a discovery-based learning experience, encouraging them to predict the outcome of each construction step before revealing the result. Students benefit from developing a deeper conceptual understanding that goes beyond rote memorization of formulas, gaining insight into why the double angle cosine formula holds true through direct geometric evidence.