A Proof of the Difference Identity for Cosine

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

Contributed by: Eric Schulz (March 2011)
Open content licensed under CC BY-NC-SA


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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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