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

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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