Cosine of x minus Sine of x/Sine Form
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Theorem
- $\cos x - \sin x = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}$
where $\sin$ denotes sine and $\cos$ denotes cosine.
Proof
| \(\ds \cos x - \sin x\) | \(=\) | \(\ds \cos x - \map \cos {\frac \pi 2 - x}\) | Cosine of Complement equals Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \, \map \sin {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \sin {\frac {x - \paren {\frac \pi 2 - x} } 2}\) | Cosine minus Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \sin \frac \pi 4 \, \map \sin {x - \frac \pi 4}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\sqrt 2 \, \map \sin {x - \frac \pi 4}\) | Sine of $\dfrac \pi 4$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 2 \, \map \sin {x - \frac \pi 4 + \pi}\) | Sine of Angle plus Straight Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}\) | simplifying |
$\blacksquare$