Cosine of Sum/Corollary

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Corollary to Cosine of Sum

$\map \cos {a - b} = \cos a \cos b + \sin a \sin b$

where $\sin$ denotes the sine and $\cos$ denotes the cosine.


Proof

\(\displaystyle \map \cos {a - b}\) \(=\) \(\displaystyle \cos a \, \map \cos {-b} - \sin a \, \map \sin {-b}\) Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle \cos a \cos b - \sin a \, \map \sin {-b}\) Cosine Function is Even
\(\displaystyle \) \(=\) \(\displaystyle \cos a \cos b + \sin a \sin b\) Sine Function is Odd

$\blacksquare$


Historical Note

The Cosine of Sum formula and its corollary were proved by François Viète in about $1579$.


Sources