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