Cosine of Sum/Proof 1

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Theorem

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


Proof

\(\ds \map \cos {a + b} + i \, \map \sin {a + b}\) \(=\) \(\ds e^{i \paren {a + b} }\) Euler's Formula
\(\ds \) \(=\) \(\ds e^{i a} e^{i b}\) Exponential of Sum
\(\ds \) \(=\) \(\ds \paren {\cos a + i \sin a} \paren {\cos b + i \sin b}\) Euler's Formula
\(\ds \) \(=\) \(\ds \paren {\cos a \cos b - \sin a \sin b} + i \paren {\sin a \cos b + \cos a \sin b}\) Complex Numbers form Field

The result follows by equating the real parts.

$\blacksquare$