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.
$\blacksquare$