Cosine of Angle plus Straight Angle/Proof 3
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Theorem
- $\map \cos {x + \pi} = -\cos x$
Proof
| \(\ds \map \cos {x + \pi}\) | \(=\) | \(\ds \frac 1 2 \paren {e^{i \paren {x + \pi} } + e^{-i \paren {x + \pi} } }\) | Euler's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {e^{i x} e^{i \pi} + e^{-i x} e^{-i \pi} }\) | Exponential of Sum: Complex Numbers | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \paren {-e^{i x} - e^{-i x} }\) | Euler's Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 2 \paren {e^{i x} + e^{-i x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\cos x\) | Euler's Cosine Identity |
$\blacksquare$