A Proof of the Difference Identity for Cosine

The key observation in the proof of the difference identity for cosine can be shown and investigated with this Demonstration.

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Difference Formula for Cosine
For any two angles and ,

Proof
Without loss of generality, assume that . Plot the points , , , and on the unit circle where



Since the arcs and have the same length, the line segments and must also have the same length. Experiment with the controls until this statement is understood and use it to investigate how the lengths of the two line segments change, but always remain equal to each other, for different values of and .



Finally, finish the proof by replacing , , , , and with , , , , and respectively.

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Related Curriculum Standards

US Common Core State Standards, Mathematics