Cosine of Difference

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

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

Proof 1

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


Proof 2


Consider two radii $OP$ and $OQ$ of a unit circle whose center is at the origin of a Cartesian plane.


\(\ds \angle xOP\) \(=\) \(\ds B\)
\(\ds \angle xOQ\) \(=\) \(\ds A\)

Then the coordinates of $P$ and $Q$ are given by:

\(\ds P\) \(=\) \(\ds \tuple {\cos B, \sin B}\)
\(\ds Q\) \(=\) \(\ds \tuple {\cos A, \sin A}\)


\(\ds PQ^2\) \(=\) \(\ds \paren {\cos B - \cos A}^2 + \paren {\sin B - \sin A}^2\)
\(\ds \) \(=\) \(\ds \cos^2 B - 2 \cos A \cos B + \cos^2 A + \sin^2 B - 2 \sin A \sin B + \sin^2 A\) multiplying out
\(\ds \) \(=\) \(\ds 2 - 2 \cos A \cos B - 2 \sin A \sin B\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds 1 + 1 - 2 \map \cos {A - B}\) Law of Cosines, as $\angle POQ = A - B$
\(\ds \leadsto \ \ \) \(\ds \map \cos {A - B}\) \(=\) \(\ds \cos A \cos B + \sin A \sin B\) simplifying


Also see

Historical Note

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