Use vector methods to prove the trig identity cos (a-b) = cos a cos b + sin a sin b

by RoRi

April 28, 2016

Prove the trig identity

$\cos (a-b) = \cos a \cos b + \sin a \sin b$

by taking the dot product of the vectors $(\cos a, \sin a)$ and $(\cos b, \sin b)$ .

Proof.
The angle between the vectors $(\cos a, \sin a)$ and $(\cos b, \sin b)$ is given by

$\cos (a-b) = \frac{(\cos a, \sin a) \cdot(\cos b, \sin b)}{1} = \cos a \cos b + \sin a \sin b. \qquad \blacksquare$

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