Cosine of Sum/Proof 3

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Theorem

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


Proof

\(\ds \cos a \cos b - \sin a \sin b\) \(=\) \(\ds \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} + e^{-i b} } 2} - \sin a \sin b\) Euler's Cosine Identity
\(\ds \) \(=\) \(\ds \paren {\frac {e^{i a} + e^{-i a} } 2} \paren {\frac {e^{i b} + e^{-i b} } 2} - \paren {\frac {e^{i a} - e^{-i a} } {2 i} } \paren {\frac {e^{i b} - e^{-i b} } {2 i} }\) Euler's Sine Identity
\(\ds \) \(=\) \(\ds \frac {e^{i a} e^{i b} + e^{i a} e^{-i b} + e^{-i a} e^{i b} + e^{-i a} e^{-i b} } 4\)
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \frac {e^{i a} e^{i b} - e^{i a} e^{-i b} - e^{-i a} e^{i b} + e^{-i a} e^{-i b} } {4 i^2}\)
\(\ds \) \(=\) \(\ds \frac {e^{i a} e^{i b} + e^{i a} e^{-i b} + e^{-i a} e^{i b} + e^{-i a} e^{-i b} } 4\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds \frac {e^{i a} e^{i b} - e^{i a} e^{-i b} - e^{-i a} e^{i b} + e^{-i a} e^{-i b} } 4\) as $i^2 = -1$
\(\ds \) \(=\) \(\ds \frac {e^{i a} e^{i b} + e^{-ia} e^{-ib} } 2\)
\(\ds \) \(=\) \(\ds \frac {e^{i \paren {a + b} } + e^{-i \paren {a + b} } } 2\) Exponential of Sum
\(\ds \) \(=\) \(\ds \map \cos {a + b}\) Euler's Cosine Identity

$\blacksquare$